The effective action of Double Field Theory

We perform a generalized Scherk-Schwarz dimensional reduction of Double Field Theory on a twisted double torus. The four dimensional effective action is shown to exactly reproduce the bosonic electric sector of gauged N = 4 supergravity. We present explicit expressions for the gaugings in terms of the twists, and analyze the associated string backgrounds. This framework provides a higher dimensional origin of the gaugings in terms of generalized fluxes.


Introduction
Flux compactifications of string theory [1] lead to gauged supergravities [2], providing an efficient mechanism of moduli stabilization and spontaneous supersymmetry breaking. An intriguing puzzle is that gauged supergravity contains more gaugings than those that can be reached through geometric compactifications of the different string effective actions or string supergravities. The presence of additional (non-geometric) gaugings suggests that some features of string compactifications have not yet been properly taken into account.
The missing gaugings could be obtained by invoking U-duality arguments at the level of the dimensionally reduced effective actions [3]- [6]. This approach corresponds to the dotted (blue) path in Figure 1. Although efficient in generating the full set of gaugings, this procedure presents the disadvantage of leaving many of them unexplained in terms of a higher dimensional origin. For instance, enforcing T-duality invariance requires the introduction of non-geometric fluxes. This can be seen by comparing the effective four dimensional superpotentials arising in orientifold compactifications of type IIA and IIB ten dimensional supergravity actions, where a broader class of internal spaces is needed to have a geometric interpretation.
The concepts of Generalized Complex Geometry [7]- [9], non-geometry [10,11] and double geometry [12]- [18] were proposed as suitable frameworks to deal with this situation (see also [19,20] for extensions to M-theory). By treating the symmetries of the NSNS antisymmetric tensor field and diffeomorphisms on an equal footing, some of the properties of T-duality are naturally incorporated. Inspired by this approach, a way to obtain the usual contribution of the locally geometric Q-flux [21] to the four-dimensional effective action, from a ten dimensional one, was recently proposed in [22].
On the other hand, there have also been attempts to promote U-duality to a symmetry at the level of the higher dimensional supergravity. For T-duality this was successfully achieved by Double Field Theory (DFT), originally formulated in [23]- [26] and extended in [27]- [37]. This attempt is represented in the figure with the dashed (red) arrow. The idea, which we review in Section 2, was to introduce additional coordinates to the standard space-time embedding of closed strings in toroidal backgrounds, dual to winding. In DFT the fields depend on both sets of coordinates and this gives rise to a 2D-dimensional theory formulated on a double space. The stringy nature of the theory is manifested in the fact that DFT is T-duality invariant, so it promotes a string duality to a symmetry.
In this work we follow the path indicated by the two solid (green) arrows in Figure 1. new degrees of freedom present in DFT as the origin of the missing (non-geometric) gaugings. The computation, for an arbitrary number of dimensions, is performed in Section 3. The reduced theory is a generalization of that found in [39]. Interestingly enough, it includes geometric fluxes as well as the locally (but not globally) geometric fluxes Q and the locally non-geometric fluxes R. Besides, it inherits a remnant gauge symmetry from DFT based on the C-bracket, and a (would be) global O(d, d) invariance which is broken by the fluxes.
In Section 4 we show that the four dimensional effective action with six internal dimensions is dual to the bosonic electric sector of gauged N = 4 supergravity, as formulated in [40]. This corresponds to the second solid (green) arrow. In order to recover the full set of gaugings, we show how to generalize the Scherk-Schwarz reduction to include the fundamental fluxes ξ +M . Additionally, we enhance the O(D, D) symmetry of DFT to O(D, D + N) by adding N generalized vector fields along the lines of [41] and [28].
Section 5 is devoted to the analysis of the string (non)geometric backgrounds and the fluxes they give rise to. We show that the internal space can be thought of as a twisted double torus. Alternative interpretations of our results from the point of view of superstring compactifications are discussed. In particular we show that the resulting effective theory can be seen either as a bosonic sector of heterotic string compactification, extended by T-duality [3,6], or as a bosonic sector of a Type IIB orientifold compactification [5,42].
Finally, the conclusions in Section 6 contain a summary of our results. We include In this section we present a brief review of the generalized metric formulation of DFT, mainly to exhibit the results that we will use throughout the paper. The notation basically coincides with the standard conventions, but we put a hat on all fields and indices to facilitate the reading of the forthcoming sections.
The building block of the theory is the generalized 2D × 2D metriĉ constructed out of the D-dimensional metricĝμν(Xμ,Xμ) and the D-dimensional 2-form fieldBμν(Xμ,Xμ). This generalized metric is an element of O(D, D), so it preserves the metric of the group that we take of the following form In addition, the model also contains an O(D, D) invariant dilaton The background independent formulation of DFT is defined by an action that can be written in a compact form (up to total derivatives) in terms of a generalized Ricci scalar as where R is defined by When the coordinates dual to winding are frozen, i.e. ∂X • = 0, this action reduces to the standard NS-NS bosonic sector of the D-dimensional string supergravities. The theory is constrained by a world-sheet level-matching condition that can be written as For further insight on DFT we refer to the original works [23]- [26] and to their many extensions [27]- [37].

Scherk-Schwarz dimensional reduction
In this section we perform the twisted dimensional reduction of DFT and compute the effective action. The starting point for the compactification procedure is the generalized 2D-dimensional double space on which DFT is defined. We identify n of its coordinates with space-time coordinates and other n with their duals. The first step is to compactify this double space-time on a double torus of vanishing dual radius, so that the usual space-time is naturally decompactified. We are therefore effectively left with a (n + 2d)−dimensional space (D = n + d) that we compactify on a twisted T d,d torus to obtain the n-dimensional effective theory of DFT.
Dual coordinates areỹ α and we usually write Y A = (ỹ a , y a ).
As we have seen, the degrees of freedom in DFT are represented by an invariant dilaton d defined in (2.3) and a generalized 2D × 2D metric (2.1), namelŷ This metric can be constructed out of a vielbein with triangular 1 form, i.e.
We have defined the D-dimensional fields that parameterize the generalized metric, namely the Kalb-Ramond fieldB and the D-beinê associated to the metriĉ These can then be written as usual in terms of the compactified degrees of freedom aŝ which in terms of the vielbein readŝ In triangular gauge (3.7), equation (3.12) can be rewritten as Parameterizing the twist in terms of a diffeomorphismẽ(Y) and B-transformationsB(Y), we find that the ansatz gives the following D-bein and Kalb-Ramond D-dimensional fieldŝ Following the organizing principle of [38], we impose that fields with only tangent space-time indices are independent of the internal coordinates and this implies the follow- where e(Y) and B(Y) are d ×d matrices and W (Y) is a d ×n matrix. By explicitly writing the dependence on the internal coordinates for the fields of the effective n-dimensional theory, we have Note that W involves space-time indices, so the ansatz produces a twist in space-time explicitly breaking Lorentz invariance. Therefore, from now on we take W = 0.
Due to the length of the computation, we show how to obtain the effective action in two steps. In the first one, we plug the ansatz (3.11) into the action of DFT (2.4). As usual, to have simple gauge transformation properties, the reduced degrees of freedom require the following redefinitions in terms of the original higher dimensional fields After some algebra, and using (2.6), we obtain 2 The dilaton Φ(x) is given by e −2Φ(x) = |detê(x)|Ṽ V I e −2φ(x) , whereṼ stands for the volume of the dual space-time and we have defined the invariant internal volume The information on the twist in (3.19) becomes manifest only through the appearance of fluxes F ABC , which are given by and, being completely antisymmetric, they belong to the 2d(2d − 1)(2d − 2)/3! repre- With this aim it appears convenient to specify the relation between the hatted generalized 2 Since the twist matrix is not necessarily globally well defined, gauge invariance of the original action (2.4) requires the additional constraint ∂ M (U −1 ) M A = 0, which we assume from now on. metric and the physical fields. Namelŷ where the physical degrees of freedom are encoded as : There are d(d + 1)/2 + d(d − 1)/2 real scalars in total, each coming from the internal components of the D-dimensional metric and B-field, respectively. In addition there is also the dilaton Φ.
• Tensors are given by the n-dimensional 2-form field B µν and the n-dimensional metric g µν .
After some algebra, the effective action can be finally written in a standard form as (see the set of equations (3.22)) Here R is the n-dimensional Ricci scalar, and we have defined the field strengths as and a covariant derivative for scalars as The effective theory ( and, when evaluated on the C-bracket (2.11), the following effective algebra is induced where the structure constants are given by the fluxes (3.20). In terms of the gauge parameter ξ A (x) the fields transform as The twists define the infinitesimal generators of this symmetry group, namely which satisfy the following C-algebra The Jacobiator associated to the C-bracket does not automatically vanish, so it is not a priori evident whether the fluxes satisfy Jacobi identities or not. To answer this question we note that the C-bracket differs from the D-bracket (which, although non skewsymmetric, does satisfy Jacobi) by a total derivative [26] [ When particularly evaluated on the twists U(Y), the last term vanishes restoring anti-symmetry and leading to a vanishing Jacobiator. We thus conclude that the fluxes must be constrained by the usual Jacobi identities We begin by reviewing this theory in the formulation of Schon and Weidner [40] and then establish the precise correspondence. The notation used in [40] differs from ours, and the relation between both conventions is specified in Table 1 and in equation (4.13) below.

Review of gauged N = 4 supergravity
The 4-dimensional ungauged N = 4 supergravity has a global symmetry G = SL(2) × O(6, 6 + N), whose maximal compact subgroup on-shell as a gauge symmetry of the theory. The part of its field content that will be relevant for our discussion is arranged in a gravity multiplet and 6 + N vector multiplets involving the following fields: • The gravity multiplet contains the metric g µν , six vectors B mµ (m = 1, ..., 6) and one complex scalar τ parameterizing the coset SU(2)/U(1). It is useful to define In addition, it contains four gravitinos ψ i µ and four spin-1/2 fermions ψ i (i = 1, ..., 4). The fermions are singlets under G and rotate among each other through SU(4) ≈ O(6) ∈ K.
• Each vector multiplet contains one vector A a µ (a labels the vector multiplet), six real scalars and four spin-1/2 gauginos λ i a . Again, the index i = 1, ..., 4 is a gauge index rotated by O(6) ∈ K, and the vector multiplets are rotated among each other by O(6 + N) ∈ K. All the 6 × (6 + N) real scalars parameterize the coset As for τ , one can construct a K-invariant scalar matrix Finally, the 6 vectors of the gravity multiplet together with the 6 + N vectors of the vector multiplets combine to form the electric vector field in the fundamental of O(6, 6 + N) In the description of [40], the ungauged theory contains the metric g µν , electric vector When these gaugings are turned on, some of the fields that were absent off-shell in the ungauged theory are now present.
The bosonic Lagrangian is a sum of three terms L bos = L kin + L pot + L top given by where the dots involve terms containing magnetic gaugings f −M N P and ξ −M . The covariant derivatives are defined by the field strength by and also the following combinations of gaugings were defined Finally, we mention that the gaugings must satisfy the following consistency quadratic

Comparison with the effective action of DFT
Having reviewed the on-shell formulation of N = 4 gauged supergravity, let us proceed to compare it with (3.25) in the particular case n = 4, d = 6. First we note that since the effective action of DFT has no SL(2) symmetry mixing electric with magnetic sectors, we can only hope to reproduce one of these sectors, which we take to be the electric one.
Since the vectors in the effective action of DFT arise only from the metric and B-field, we must set N = 0. We then show how to relax this restriction and finally, by gauging the rescaling symmetry, we propose a way to obtain the gaugings ξ +M .
Note that the O(6, 6) metrics employed in (3.21) and (4.2) differ, the relation between them being (12) . There is no other source of deformations so far, so we must also set ξ +M = 0 (the inclusion of the full set of electric gaugings is discussed in the forthcoming subsections).
To perform the comparison, the effective action of DFT (3.25) must first be taken to the Einstein frame. The transformation g µν → e 2Φ g µν gives Then, one should identify Effective DFT action Gauged N = 4 sugra Table 1 and this guarantees that both actions and also their respective constraints coincide. The following subtleties must be taken into account when performing the comparison: 1. Due to quadratic constraints (4.12), when f −M N P = ξ −M = ξ +M = 0, the topological terms vanish L top = 0. To see this, note that (4.15) 2. The term g 2 f +M N P f +M N P in L pot can be explicitly written in terms of the twist, by using (3.20), as and therefore it vanishes due to the level matching condition. This in turn implies that the effective action (4.14) is in fact a truncation of an N = 8 theory to N = 4 [42,43], so only a subset of gauged N = 4 supergravities can be reached through Scherk-Schwarz compactification from DFT.
3. To make the equivalence evident, the two-form B µν must be dualized in the effective action of DFT to give an equivalent theory in terms of a scalar field λ ≡ 2Re(τ ).
Introducing λ as a Lagrange multiplier to enforce ǫ σµνρ ∂ σ ∂ µ B νρ = 0, one defines the following action for B µν Here G µνρ must be eliminated through its equation of motion which replaced in S B gives The first term contributes to the kinetic term for the gravity scalars, and the second one coincides with the last term in L kin upon use of quadratic constraints.

Including vectors
We showed above that the effective action of DFT matches the bosonic electric sector of gauged N = 4 supergravity when the number of vector multiplets is six, i.e. N = 0, and ξ +M = 0. Here we show how to relax the first restriction.
To increase the number of electric vector fields, we are inspired by the extension of DFT to the abelian heterotic string performed in [28], where the 2D coordinates Xμ,Xμ,μ = (4.18) We formally keep the action and the form of the gauge transformations but with respect to the enlarged generalized metricĤMN introduced in [28], namely where the indices of the gauge group i, j, . . . are raised and lowered with δ ij and The generalized vielbein producingĤMN = EÂMηÂBÊBN iŝ The compactification procedure closely follows [39]. The metric and B fields in terms of the compactified degrees of freedom take the form (3.10), where now 22) and the vector fields are Generalizing the procedure of the previous section, we define the reduced gauge multiplet as 24) and the reduced generalized metric as where The global symmetry to be twisted is now O(D, D + N) and we extend the ansatz (3.11) for the generalized metric aŝ where Z = (Y, z) andÛ ∈ O(D, D + N). In terms of the vielbein, it readŝ which can be rewritten in matricial form as Parameterizing the twist in terms of diffeomorphismsẽ,B andṼ transformations, namelŷ the ansatz gives the following D-dimensional gauge fieldŝ

Rescaling symmetry gaugings
In this section we propose a way to obtain the gaugings ξ +M in the (2, 12) representations of SL(2) × O(6, 6). For simplicity here we restrict to the N = 0 case, but the results can be extended if heterotic vector multiplets are introduced as in the previous subsection and their rescaling taken into account. To facilitate the presentation, in this subsection we turn off the gaugings in the (2, 220).
In [44], a rescaling symmetry of the NS-NS sector of the heterotic theory was twisted, and it was shown that this gives rise to the gaugings ξ +m with m = 1, . . . 6. This symmetry acts by rescaling the metric and B-field, and by shifting the dilaton, and it is explicitly broken in DFT due to the dependence on the dual coordinates. However, at a formal level DFT is invariant under a global symmetry given bŷ  While the former (4.33) corresponds to the case analyzed in [44], the later is only possible due to the doubling of the coordinates, and symmetry arguments suggest that they should give rise to the ξ m + gaugings. Here we use the covariant notation (4.31), but one should keep in mind that the extension of the symmetry is only possible if covariance is broken through either (4.33) or (4.34).
The compactification procedure is equivalent to the one before. One transforms the fields and makes the transformations carry the internal dependencê (4.35) The ± depends on whether (4.33) or (4.34) is chosen.
The comparison of the effective action with gauged N = 4 supergravity is not straightforward, and one must use the dualizations of [44]. Instead of dualizing the B-field into the axion of [40], one must integrate the A M − µ fields out and use the local gauge invariance under (axionic) rescaling shifts to gauge away the axion Imτ = 0. This yields the following equivalent formulation of gauged N = 4 supergravity when only ξ + gaugings are turned on ) The identifications are those of the previous section plus B ++ µν ↔ B µν and the gaugings are originated from the following derivatives (4.41) Equations (4.33)-(4.34) imply that the constraints ξ M + ξ +M = 0 are automatically satisfied.

(Non)geometric string backgrounds
In this Section we show that the generalized Scherk-Schwarz mechanism can be interpreted as a reduction on a twisted double torus. The string backgrounds associated to different twists are analyzed and their connection to standard (dual) fluxes is discussed.

Reduction on a twisted double torus
The dependence of the generalized metric and its vielbein on the internal coordinates Y M was introduced in (3.11) as In the limit in which the scalar matrix approaches the identity H AB (x) → δ AB , the generalized metric takes the form It is useful to introduce the 1-forms When the double space is untwisted, these 1-forms are simply the differentials dY A , which are globally well defined and covariantly constant with respect to the standard derivative d. The Scherk-Schwarz twisted torus [38] has instead globally well defined 1-forms, Γ a = U a α dy α , which are covariantly constant with respect to a derivative that includes a constant non-vanishing Levi-Civita connection ω a b . This becomes manifest through the Maurer-Cartan equation Interestingly, f a bc := −ω a [bc] can be defined through the Lie bracket that in turn defines the gauge symmetry of the effective action, and in fact, the connection is precisely given by the fluxes. The Scherk-Schwarz twisted torus can be conceived as the manifold of a group, the components of f being its structure constants. Such group is the gauge group of the effective action, where the parameters f appear as gaugings.
We have seen already that many of these features are preserved in our scenario. For instance, the standard Lie bracket is replaced by the C-bracket which determines the gauge invariance of the effective action. However, it is not a priori clear what is the relation between the fluxes F ABC and the connection and torsion of the double twisted torus.
The proper formalism to deal with in the context of double field theory was developed in [36], generalizing the frame-like geometrical formalism introduced by Siegel [45]. It is useful to recall the following definition of the 2-form torsion where L V is the Lie derivative of vector fields and the upper index ∇ means one must change the partial derivative by the covariant one.
So, we define the generalized torsion in the context of double field theory as in (5.5), where the label a is replaced by A and the standard Lie bracket is replaced by the Cbracket [36] (see [46] for a discussion of torsion in the context of Generalized Geometry).
Using the Lie derivative (2.9) and the covariant derivative where ω A B represents the generalized spin connection, and imposing the metricity condition (which implies ω AB = −ω BA ) one gets

Generalized fluxes and superstring compactifications
The low-energy effective theories arising in standard compactifications of string theory on manifolds with reduced structure or orbifolds are gauged supergravities in which only a subset of gaugings can be obtained. Those gaugings whose origin from a 10-dimensional supergravity has not been identified are dubbed non-geometric. From a stringy perspective, the main motivation for introducing these fluxes is duality. Interestingly, being invariant under T-duality transformations, DFT provides a higher dimensional framework in which all fluxes related to the antisymmetric field strength H-flux through T-duality appear on an equal footing.
We have shown in Section 4 that the compactified theory can be identified with the bosonic electric sector of N = 4 gauged supergravity with broken global symmetry group G = SL(2, R) × O(6, 6 + N). From a stringy perspective, the N = 4 supergravity theory could arise, for instance, from the compactification of D = 10 heterotic string [39], but also from compactifications of Type II either on toroidal orientifolds [5], [47] or on SU(2)structure manifolds [48]. The global group G has different interpretations in each case (see [6] for details). Bρν in (3.6) is the Kalb-Ramond field of the heterotic string, we see from (4.17) that τ = λ + ie −2φ would indeed be the heterotic axio-dilaton field. In what follows we will pursue this interpretation, but we will comment at the end of this section that a Type IIB orientifold interpretation is seemingly possible.
We now address the issue of the 2d-dimensional origin of the fluxes. Starting from their definition (3.20): we parameterize the internal 2d-bein in the most general form (see Appendix A.1) One can then define the relation between the tensor F ABC and the standard set of NS-NS string fluxes of [3]. Defining for simplicityã ≡ a + d, we set The different fluxes take the following form in terms of this parameterization , (5.14) where underlined indices must not be antisymmetrized.
It is interesting to analyze the dependence of the fluxes in different limits of interest: • Setting β = 0 and taking B, e to depend only on base coordinates, the H-flux takes the geometric form and it can be seen from (5.18) that they leave the gaugings invariant.
Notice that, from the first and third terms in (5.16) and keeping only ordinary derivatives, we obtain Q ab c = β ra β bs H rsc + .. reproducing the expression found in [9] (see also [22]).
Having made contact with string fluxes, we now define the subgroup of O(d, d) transformations whose elements correspond to T-dualities, namely where the index α denotes the direction in which the T-duality is performed. As expected, these elements satisfy the following properties The T-duality operators (5.25) lead to the Buscher rules introduced in [49]. We have seen in (2.7) how the generalized metric transforms under these symmetries. Equivalently, one can define a unified tensor E = g + B [50], which transforms as Taking the O(d, d) transformation to be a T-duality defined in (5.25) in the y α direction, one obtains the following transformations for E which decomposed into symmetric and anti-symmetric parts lead to the well-known Buscher rules Defined like this, the standard T-duality chain [3] holds and when the Jacobi identities (3.35) are written in terms of the fluxes (5.13), the following results of [3] are exactly recovered  which are the Jacobi identities derived from the heterotic algebra found in [39], after adjusting some normalization factors.

Type I interpretation
As we mentioned at the beginning of this section, the effective DFT four dimensional theory can be equally well interpreted in terms of a Type IIB orientifold compactification.
For instance, if instead of the identifications (5.13) we had used where now F abc is the RR 3-form flux of Type IIB compactification with O9 planes, ω a bc is a geometric flux and for simplicity we turn off the non-geometric fluxes. In this case, the Jacobi identities would read where the first equation is the usual constraint on geometric fluxes, the second equation is the requirement of absence of Freed-Witten anomalies and the third one is a tadpole cancelation equation [5] if vector fluxes are now interpreted as D9 magnetic fluxes.
The full set of Type IIB generalized fluxes can be identified (see for instance [5] in an O3 notation). For bulk fluxes, they correspond to a definite SL(2, R) spin (say plus in the notation of Section 4) in the (2, 220) representation of SL(2, R) × O(6, 6) B . The opposite spin would correspond to S-dual fluxes.
From the point of view of the starting DFT, the original fields have now a different interpretation. Namely, theBρν field entering the generalized metricĤMN in (2.1) is such that in the ten dimensional effective field theory it reproduces the two form RR field.
Namely, the DFT effective theory in ten space-time dimensions now reads where F is the RR 3-form field strength.

Conclusions
We the formulation of [40], and the precise dualizations required to make the equivalence explicit can be found in Section 4. Furthermore, the global symmetry group of the effective action was enhanced to O(d, d + N), by considering, as a starting point, the abelian heterotic extension of DFT formulated in [28]. In addition, the gauging of a generalization of the rescaling symmetry considered in [44] was proposed as an origin for the ξ M + fluxes. Altogether, our results imply that all the electric gaugings of N = 4 gauged supergravity can be reached from DFT.
It is worth mentioning that throughout the paper a strong constraint, as discussed in section 2, was assumed. This is needed for gauge invariance of the DFT original action.
Interestingly enough the Scherk-Schwarz like dimensional reduction to go from DFT action (2.4) to the effective action (3.25) does not require further use of the strong constraint 3 .
This also applies, in particular, for Jacobi identities implying the "N = 4 quadratic constraints", needed in gauged supergravities for the closure of the gauge transformations.
A similar observation was pointed out in [52] (see also [27]). It would then be interesting which, restricted to h −1 = e T ± e ± , leads to the usual form of the generalized metric Given any elementÕ ∈ K of the form Note also that the form of the vielbein is now and we can perform a K transformation with O ± acting on e ± to set e + = e − = (e −1 ) T , and take E to triangular form We have then managed to write it as a product of diffeomorphisms and B-transformations.
Any β-transformation would take it away from the triangular gauge, to a form parameterizable as in (A.7), so it can always be brought to triangular form through a K-rotation. It is worth emphasizing that, although the K-transformation seems to restore the effect of the β-transformation, the restoration is only structural, and in many cases it is performed at the expense of ending with non-geometric backgrounds h and b. In this paper we have chosen the triangular gauge to do the computations, but the results are finally written in a covariant way.

A.2 A single flux example
Here we would like to consider simple backgrounds, on which we apply T-dualities and track how the obtained (non)geometries are encoded in the double torus. For simplicity we turn on a unique flux for which all the consistency equations are automatically satisfied.
We closely follow the procedure of [51], where it is analyzed how the different fluxes twist the double torus through successive T-dualizations and how geometric and non-geometric backgrounds are obtained after projecting the different twisted double tori to a base.
Although the definition of flux is different here than in [51], we show that the conclusions remain unaltered.
Now we can compute the IIB kahler modulus and its shift under y 1 → y 1 + 1 as .26) and this is again a fiber of T 2 1 shifting the kahler modulus as we move in the direction y 1 . Although this background is locally described in terms of a metric and a B-field, upon going around a cycle y 1 → y 1 + 1, the metric and B-field mix through an S ∈ O(2, 2) transformation OU Q (y 1 + 1)S = OU Q (y 1 ), defined by where we have already projected to the base. The fact that under a cycle the metric and B-field mix implies that this background is not globally well defined. However, as seen, locally it is described by a metric and a B-field. Therefore, these backgrounds are said to be locally geometric, but globally non-geometric.

R-flux
Finally, we apply the remaining T-duality in the direction y 1 to obtain a background with flux R 123 = 3N where and the monodromies (y 4 , y 2 , y 3 ) → (y 4 + 1, y 2 + Ny 6 , y 3 − Ny 5 ) , y 1 → y 1 + 1 , (y 5 , y 1 , y 3 ) → (y 5 + 1, y 1 − Ny 6 , y 3 + Ny 4 ) , y 2 → y 2 + 1 , (y 6 , y 1 , y 2 ) → (y 6 + 1, y 1 + Ny 5 , y 2 − Ny 4 ) , y 3 → y 3 + 1 . (A.30) The matrix U R reduces to the identity when projected to the base, and must then be associated to a trivial torus with vanishing B-field. This can be understood if we take into account that in every step of the T-duality procedure we project out the new dual coordinates, and this successively reduces the flux by units of N. Since we started from a background with H 123 = 3N and performed three T-dualities, we have ended after three successive projections with a background having R 123 = 0. In summary, T-dualities incorporate dual fluxes into the game which we are forced to project out if we want to make contact with some string background geometry. In this example, this can be done for H, ω and Q but not for R which then receives the name of locally non-geometric flux.