The topology of Double Field Theory

We describe the doubled space of Double Field Theory as a group manifold G with an arbitrary generalized metric. Local information from the latter is not relevant to our discussion and so G only captures the topology of the doubled space. Strong Constraint solutions are maximal isotropic submanifold M in G. We construct them and their Generalized Geometry in Double Field Theory on Group Manifolds. In general, G admits different physical subspace M which are Poisson-Lie T-dual to each other. By studying two examples, we reproduce the topology changes induced by T-duality with non-trivial H-flux which were discussed by Bouwknegt, Evslin and Mathai [1].


Introduction and summary
Dualities play a crucial role in our current understanding of string theory. S-and Tduality connect the five perturbative superstring theories in an intriguing way and led to the discovery of M-theory [2,3]. Due their complexity, usually only their low-energy effective actions are studied. Unfortunately, T-duality is not directly manifest at this level. In the NS/NS sector of type IIA/B on a d-dimensional torus, it is mediated by the Buscher rules [4]. They link dual target spaces in complicated, non-linear way.
The major objective of Double Field Theory 1 [5][6][7][8] (for reviews see [9,10]) is to make the T-duality group O(d, d, Z) of a d-dimensional torus manifest. To this end, the theory is formulated on a doubled space with momentum and winding coordinates. There is a clear distinction between weakly and strongly constrained DFT. While the former is only valid on a tours, the latter works for arbitrary target spaces. It imposes the Strong Constraint (SC) which restricts all fields to depend on a d-dimensional physical space only. In doing so all supergravity backgrounds are accessible in DFT. At the same time the notion of the JHEP04(2018)128 doubled space becomes more subtle. Depending on the target space topology there are in general much less winding modes than on a torus. Hence, one is inclined to ask: what is the significance of the doubled space in the strongly constraint theory? Does it still captures relevant information about T-duality or is it just a convenient bookkeeping device?
Currently, the doubled space of DFT is argued not to be a differentiable manifold [10]. 2 In a single patch this does not cause problems because one is always able to solve the SC by choosing d out of the 2d coordinates for the physical space. All fields in the theory only depend on them. Gluing different patches together is more involved. Finite generalized diffeomorphisms are the natural candidates for transition functions and therefore were intensively study recently [15][16][17][18][19]. But they rely on solving the SC first and patching afterwards which results in subtleties for target spaces with H-flux in a non-trivial cohomology class [20,21]. Hence going beyond toroidal backgrounds, the global structure of the doubled space is not fully understood yet. Especially in the context of T-duality this is a pity. In a coarse approximation, it exchanges momentum and winding modes which crucially depend on the topology of the target space. Thus, global properties have to play a central role. Indeed, T-duality can induce topology changes of the target space [1,22,23] which has important implications for brane and string charges. This paper presents a different approach to the doubled space. We describe it as a 2d-dimensional Lie group G which admits an embedding into O(d,d). Doubled geometries of this kind [24][25][26][27] were originally introduced by Hull and Reid-Edwards. They have a natural interpretation in lower dimensional gauged supergravities which arise from Scherk-Schwarz compactifications [28][29][30][31][32]. However, they did not affect the perspective on the doubled space in DFT until recently when DFT WZW [33][34][35] was proposed. It is a formulation which is derived from a Wess-Zumino-Witten model and implements a non-trivial H-flux by construction. Like in DFT, the doubled space G is equipped with an arbitrary generalized metric. For the following discussions this metric is irrelevant. Only the topology of G matters. By solving the SC of DFT WZW , one embeds the theory's physical subspace M , which is restricted to have the topology of a coset space, in G. In general, there are different SC solutions and each of them gives rise to a dual background. Two explicit examples are studied in this paper. They reproduce the topology changes presented in [1]. Furthermore, this approach allows to patch the doubled space with standard diffeomorphisms instead of the finite generalized ones mentioned above. It is known that the latter are insufficient in implementing the full O(d,d) and more general transformations are required. Normally, they are introduced by hand. But they also admit a geometric interpretation [14]. Our approach exploits this geometric perspective and so circumvents the problems outlined above.
A SC solution identifies a d-dimensional physical subspace M in G. To obtain it, we choose a maximal isotropic subgroup H ⊂ G and consider the coset M = G/H. In general one can choose between different subgroups. Each one gives rise to a T-dual solution. The same prescription is used to define Poisson-Lie T-duality [36][37][38]. Of course T-duality also imposes additional constraints on the generalized metric and dilaton. We do not discuss JHEP04(2018)128 these restrictions in this paper, because we are merely interested in metric independent statements. Furthermore, the notion of T-duality used here is Poisson-Lie T-duality [36]. It includes abelian and non-abelian T-duality as special cases and is currently the most general notion of T-duality available. There might be the concern that Poisson-Lie Tduality is not a full symmetry of string theory. In contrast to abelian T-duality, it does not hold to all orders in the string coupling g s and the string tension α . Still, at the two derivative level (leading order α ) on which we are discussing DFT here, it has exactly the same properties as abelian T-duality. In particular already the canonical example of the T-duality chain for the torus with H-flux, which we present in section 4.1, is an example of Poisson-Lie T-duality bases on the non-abelian Drinfeld double cso (1,0,3).
Only fixing H is not sufficient to construct a complete SC solution. Additionally, we need a map σ i : U i → G in each patch U i ⊂ M . Fixing this map is the main challenge we have to face. To this end, G is interpreted as a H-principal bundle π : G → M with a connection one-form ω splitting its tangent space into a horizontal and a vertical part T G = HG⊕V G. We choose ω in a particular way such that HG contains all tangent vectors of the d directions in G solving the SC. Finally, we have to choose a σ i whose differential map only has HG as image, but not V G. Equivalently, we have to fulfill the constraint which requires a locally vanishing gauge potential A i on M . This procedure fixes both σ i and ω. First, we choose ω such that the field strength for all A i in all patches vanishes. In this case the gauge potential is locally a pure gauge and we can set it locally to zero by an appropriate gauge transformation, fixing σ i at the same time. If the bundle is not trivial, we can not set A i to zero globally. However as we explain in section 3.1, this is not required to find a global solution of the SC. It is totally sufficient to have a flat connection. We further show that the connection one-form is isomorphic to a pure spinor which specifies the polarization on the group manifold. Figure 1 illustrates the relation between G and M . All remaining unphysical directions are represented by the gray line in this figure. Their tangent vectors span V G and are isomorphic to elements in the Lie algebra h of H. By defining an isomorphism η : h → T * M , we are able to identify V σ i p G with the cotangent space T * p M at each point p ∈ U i . Finally, we obtain the exact sequence where assigns a fundamental vector field to each element of h. In conjunction with the generalized Lie derivative of DFT WZW , this sequence represents an exact Courant algebroid. We use its maps to define a generalized frame fieldÊ A . It is an element of O(d,d) and allows to pull all fields on the group manifold to the generalized tangent space T * M ⊕ T M . Thus, for each solution of the SC we obtain a Generalized Geometry (GG) [39,40] with a twisted Courant bracket. There are two ways how such a twist shows up [41]. It can be realized as a twist term in the Courant bracket or as a gerbe for the B-field in the frame fieldÊ A . In the latter case, the generalized tangent space is replaced by a twisted bundle JHEP04(2018)128 Figure 1. The curved, black line represents all directions in G which solve the SC. Their tangent space forms the horizontal subspace defined by the connection one-from ω as ω(X) = 0 ∀X ∈ HG. All tangent vectors on the physical space M are mapped to HG by the differential map σ i * in the appropriate patch U i ⊂ M .
E which only looks locally like the sum of M 's tangent and co-tangent bundle. In general SC solutions in DFT WZW have both contributions. How they are distributed depends on the specific properties of the group manifold G and the chosen subgroup H. We present two explicit examples, the torus with H-flux and the three-sphere S 3 with H-flux. They implement T-duality in very different ways. For the former, we obtain the T-duality chain [42,43] H ijk (T i denotes T-duality along x i ) by choosing different, maximal isotropic subgroups H ⊂ G. All possible subgroups are classify by Lie algebra cohomology and related by O(d)×O(d) transformations. Similar results are known in the current formulation of DFT. One important difference is that the H-flux of the background splits into a gerbe and a twist contribution. This splitting is not arbitrary but completely fixed by the group manifold G=CSO(1,0,3). At the same time the solution of the SC is twisted. While in each patch one can split the 2d coordinates on G into d physical and d unphysical ones, they mix globally. T-duality acts as a change of coordinates on G. Only fields with an isometry along the T-duality direction solve the SC in both frames. We are not able to find SC solutions with R-flux.
For the S 3 with h units of H-flux the only possible subgroup of G=SO(4) is H =SO(3). As a simple Lie group its algebra h does not admit any non-trivial deformations. Its SC solution implements the complete H-flux in the twist of the Courant bracket. Despite the lack of different subgroups, one can still obtain a T-dual background by modding out a discrete subgroup which results in M =Z h \SO(4)/SO(3), the lens space L(h, 1) with one unit of H-flux, as physical space. However, we do not obtain a T 2 fibration over S 2 as described in [44]. The reason is that the CFT for the dual target space is obtained from a SU(2) WZW-model at level h by orbifolding a finite group [45,46]. As result there is only a finite number of winding modes in the twisted sector. They are not sufficient to furnish

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all Fourier modes on a circle and thus cannot be associated to the winding coordinate of a S 1 dual to the one of the S 1 Hopf fiber of S 3 .
All results obtained in this work allow to refine the statements about the status of DFT WZW compared to the traditional formulation. Locally, they are equivalent after solving the SC [34,47]. Globally, DFT WZW contains additional information about the topology of the doubled space. The remaining part of the paper is organized as follows: a compact review of the salient features of DFT WZW is given in section 2. The technique to solve the SC, we already outlined above, is developed in section 3. For symmetric spaces M , we show how such solutions can be constructed very explicitly. Section 4 presents the tours and the S 3 , both with h units of H-flux, as examples. We close with an outlook in section 5 which highlights possible applications and directions for future projects.

Review of Double Field Theory on Group Manifold
In the following we review the features of DFT WZW [33,34] which are essential to discuss non-trivial solutions of the SC. A broader review is given in [48]. The theory is formulated on a doubled space equivalent to a Lie group G with the coordinates X I (I = 1, . . . , 2d). G carries an O(d, d) structure which allows to introduce a globally defined metric η IJ with split signature. Because Lie groups are parallelizable, one is able to introduce a global generalized frame E A I ∈GL(2d) (and its inverse transpose E A I ) which is called background generalized vielbein. It is used to bring the η-metric in the canonical form In general there are different ways how to choose this canonical form. They are all related by GL(2d) transformations. In [33], it is diagonal and makes the splitting in left-and right-movers on the string worldsheet manifest. Here, we use a representation inspired by the standard formulation of DFT. Thus, flat indices A have the structure where a = 1, . . . , d. We denote the generators of G's Lie algebra g by t A and θ A is the corresponding dual one-form (θ A (t B ) = δ A B ). A particular choice for g is picked by fixing the structure coefficients They implement non-trivial background fluxes in the theory. The Lie algebra structure carries over to the flat derivatives D A = E A I ∂ I which fulfill the analog identity DFT WZW has two dynamical fields, the generalized metric H IJ and the generalized dilaton φ. Their dynamics is governed by the action [34]

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which can be expressed in terms of the generalized curvature scalar In this equation, a second covariant derivative ∇ A appears. It is connected to D A by the relation Furthermore, its action on curved indices is chosen such that it is compatible with the background generalized vielbein The DFT WZW action (2.5) is invariant under generalized diffeomorphisms. Infinitesimally, they are generated by the generalized Lie derivative However, these transformations only close into a consistent algebra if the strong constraint (SC) hold. · is a place holder for fields, parameters of generalized diffeomorphisms and arbitrary products of them. Basically, this constraint requires that all fields only depend on a ddimensional subspace M of the doubled space. Locally there is not more to add to this statement. Globally it is more subtle as this paper shows. In addition to generalized diffeomorphisms, the action and the generalized Lie derivative transform covariantly under 2d-diffeomorphisms. They are mediated by the standard Lie derivative [34] and their appearance seems natural since we started from a Lie group G as doubled space. Remember, each Lie group is a differentiable manifold. Thus, we should be able to find all structures known from differential geometry and topology. We exploit them to construct explicit solutions of the SC in the next section. This is the main difference between DFT WZW and the traditional formulation [6,8,49]. There, the doubled space does not carry any additional structures besides generalized diffeomorphisms.

JHEP04(2018)128 3 Solutions of the section condition
For the consistency of the DFT WZW the SC (2.11) plays a crucial role. It can be trivially solved by making all fields and parameters of gauge transformations constant on the group manifold. Such solutions are used in the generalized Scherk-Schwarz compactifications presented in [35]. Here, we are interested in a more general class. We construct explicit solutions by exploiting that the doubled space is a differentiable manifold which can be equipped with the structure of a H-principal bundle. In subsection 3.1, we explain how such bundles capture all data that are required to describe a global solution of the SC by embedding a physical submanifold M = G/H in G. While the generalized Lie derivative (2.10) acts on the tangent space T G of the group manifold, we use the maps defining the H-principal bundle to pull it to the generalized tangent space T * M ⊕ T M . This link between DFT and GG is known since the early days [7]. However, we will see that globally the generalized tangent bundle gets twisted as a result of the non-trivial topology of the group manifold. Capturing this twist without any reference to the generalized metric is one of the merits of the formalism presented in this paper. As we finally discuss in subsection 3.4, the SC admits in general more than one solution.

Reformulation as H-principal bundle
We now present a systematic way to find SC solutions on a group manifolds. Following [50], we first substitute its quadratic version through the equivalent linear constraint It allows us to specify a pure, 3 Spin(d,d) Majorana-Weyl (MW) spinor Λ α for all distinct solutions. MW spinors carry indices α = 1, . . . , 2 d−1 and Γ A αβ denotes the corresponding Γ-matrices defined by We assign a value to this spinor on each points of the group manifold G. In order to relate different points, we remember that G is equipped with a Lie algebra g which generates infinitesimal translations. Its generators are completely specified by the structure coefficients F AB C and can either act on the 3 A pure spinor is annihilated by the maximal number of linear independent Γ-matrices. Thus, it selects a maximally isotropic subspace as for example explain in section 3.3 of the review [41].

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representation of O(d, d). Applying the exponential map to them, one obtains group elements in these two representations denoted as g A B and g α β , respectively. Furthermore, assume we found a set of fields f with a specific coordinate dependence such that they solve the linear constraint (3.1) for a fixed Λ α . In this case there also exists another set of fields, we call them f , with a different coordinate dependence solving the linear constraint for We denote the adjoint action of a group element g on a Lie algebra element in the first equation as This property of the linear constraint (3.1) is essential for what follows. It results from the fact that the Γ-matrices transform covariant under O(d, d). By construction, G is a subgroup of this group and thus (3.1) transforms as a MW spinor and is zero for each g ∈ G.
Using this nice property of (3.1) allows us to introduce a map between group elements g and distinct solutions of the SC. In order to obtain an explicit expression for this map, we fix Λ α to an initial value Λ α 0 and ask which different values for Λ α are possible after applying (3.5) to all group elements g ∈ G. The subset of elements leaving the specific choice Λ α 0 invariant are assigned to the stabilizer subgroup H ⊂ G. This permits us to decompose each group element into and m denoting a coset representative in the left coset G/H. The given structure allows us to identify G with a H-principal bundle, whose tangent space fits into the exact sequence In the following, we fix all maps which appear in this diagram. Each element in G is parameterized by the coordinates X I . Furthermore, we assign to each coset representative m the coordinates x i and to an element h ∈ H of the subgroupxĩ. By doing so, we explicitly implement the splitting ) of the coordinates on the group manifold. In terms of these adapted coordinates the projection π(X I ) = x i (3.10)

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only chops thexĩ part of X I . From this equation, we deduce the corresponding differential map The tangent space T g G at an arbitrary point of the group manifold is linked to a Lie algebra element by the left-invariant Maurer Cartan form which is identical with the background generalized vielbein. Moreover, there exists an isomorphism between left invariant vector fields and Lie algebra elements. It has the property ω L (t A ) = t A and reads This fixes all maps in (3.8) going from left to right. The opposite way is more involved.
In order to find the remaining maps σ i and ω, first note that generators in the Lie algebra g = m ⊕ h can be split into two complement sets with t a ∈ m and t a ∈ h (3.14) if H is a maximal isotropic subgroup. Employing this splitting to the left-invariant Maurer-Cartan form (3.12) with the group element (3.7), we find A quick calculation confirms the property π * (t a ) = E ai ∂ i = 0 of the exact sequence (3.8).
Next, we take a closer look at the h-valued connection one-form ω of the H-principal bundle. It has a nice geometric interpretation as splitting the tangent space T G = HG ⊕ V G into a horizontal part HG and a vertical one V G. While the latter is spanned by the fundamental vector field (t a ) , the former is defined as the kernel of the connection. For this splitting to be consistent, the connection has to fulfill the two constraints where R g denotes right translations R h g = gh on the group manifold. We want to fix the connection in such a way that the resulting subspace HG of T G solves the linear constraint (3.1). To this end, we define a projector P m at each point m of the coset space with the important property P m t a = t a ∀t a ∈ h .

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Remember, θ A is the one-form dual to the Lie algebra generator t A define in (2.2). In order to fix a distinct solution of the SC on the complete group manifold, we extrapolate this projector from the coset G/H to G by the prescription Now we have everything we need to finally define a connection one-form with the desired properties (3.17).
Last but not least we come to the local sections σ i , which are defined on the patches U i ⊂ G/H covering the coset space. The most general local section is specified by a set ofj = 1, . . . , dim H functions fj i in each patch. It fulfills πσ i = id G/H and its differential map reads We are now able to calculate the canonical pullback of the connection one-form For more general local sections with arbitrary functions This a gauge transformation for the gauge potential A i . If we can write A 0 i as there is a particular choice of the functions fj i such that the pullback A i vanishes. An equivalent statement is that the field strength vanishes. If this happens in all patches, we found a flat connection. This is exactly the case we are interested in. Now, the sequence (3.8) becomes exact also from right to left because ωσ i * = 0. In the overlap region U i ∩ U j between two patches the local sections are connected by where t ij : U i ∩ U j → H denotes the transition functions of the bundle. Because the connection one-form is uniquely defined on G, we obtain the compatibility condition

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The condition ωσ i * = 0 only fixes the section locally. In general there is no global section (all transition functions t ij = e are the identity e ∈ H). A principal H-bundle is trivial iff it admits a global section. Otherwise, we are not able to split T G = V G ⊕ HG globally. Still, non-trivial H-bundles are not an obstruction in solving the SC globally. In order to prove this statement, assume that we already solved the SC locally in all patches. This is equivalent to find coordinates on each patch U i in which the curved version of η IJ has the form Here • is a placeholder for in general non-vanishing contributions. Coordinates in overlapping patches U i and U j are connected to each other by the transition function ϕ ij : As long as ϕ ij is restricted to Note that the connection F on this bundle was introduced as a tool to discuss solutions of the SC. There is more familiar connection on the H-principal bundle. It results from restricting the left-invariant Maurer-Cartan form to the subalgebra h. Pulling it back to T * G/H, we obtain the gauge potential where (E i ) A J denotes the background generalized vielbein in the patch i. We will see while presenting examples in section 4 that its field strength Even if the SC is solved globally. It can be used to classify the bundle in terms of its characteristic classes.

Explicit form of the connection
In this subsection we take a closer look at the projector P m which is used to fix the connection one-form ω on G. So far, we treated it as an abstract object by only requiring its elementary property (3.19). Now, we construct it explicitly.
To this end, we first relate the Γ-matrices in (3.1) to fermionic creation/annihilation operators ψ a /ψ a . They fulfill the canonical anti-commutator relations (3.33)

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If we scale them accordingly, they are equal to Γ a = √ 2ψ a and Γ a = √ 2ψ a (3.34) and reproduce the Clifford algebra (3.2). We define the pure spinor |Λ 0 from the last subsection to be the normalized vacuum state of this setup Thus, it is annihilated by In the following we suppress spinor indices and use a bra-ket notation instead. Hermitian conjugation relates creation and annihilation operators That is all we need to write down the first version of As one immediately sees, it is a trivial implementation of the property (3.19 and the even, totally anti-symmetric 2n forms B (2n) up to degree 2 d/2 . Replacing the right-hand side of (3.38) with this spinor results in Let us explain this expression. First, the constraint (3.19) fixes the B (0) contribution to be one. At most there are two annihilation operators on the left-hand side of the bra-ket. They can only compensate two creation operators on the right. Thus, all contributions B (2n) from (3.39) with n > 1 do not contribute to (3.40). This leaves us with B (2) which we rename to B. In order to simplify this result further, one replaces the Γ-matrices by the corresponding creation/annihilation operators and swaps them until all creation operators are on the left. This procedure results in the simple expression By construction, this projector has all SC solutions for Λ as its nullspace. After combining it with the left-invariant Maurer-Cartan form, we obtain the gauge potential

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Note that we suppress the index labeling the patch dependence of the section for the sake of brevity here and also in the following. For A = 0, we are able to reconstruct the solution of the SC on each point of the coset G/H in terms of the two-form This result is interesting, because it provides a shortcut to obtain a vanishing connection. Normally, we would first try to fix the two-form B in such a way that the field strength F in (3.26) vanishes. In this case A is a pure gauge and one can set it to zero by an appropriate gauge transformation in every patch. An equivalent statement is that the symmetric part of δ b a E bi E a j vanishes. This is the case iff holds. In the following, we show that η ij = 0 automatically holds if: first, we choose a coset representative of the form where U denotes a patch of the physical subspace G/H. And second, the two complement sets g = m ⊕ h spanned by the generators t a and t a fulfill the relations They render m and h to be a symmetric pair and consequently the coset G/H is a symmetric space. In order to keep the following expressions compact, it is convenient to introduce the bilinear inner product t A , t B = η AB . (3.47) In this notation, we have to show that In order to evaluate this expression, we use the Baker-Campbell-Hausdorff relation which gives rise to Furthermore, we exploit that g can be embedded into o(d, d) to identify which after several iterations allows to derive the relation

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After applying it n − m times, where n > m, we obtain Now we are able to decompose the expansion for (3.48) into the two contributions Hence, we are able to construct a flat connection for all group manifolds whose Lie algebras admit a splitting into a symmetric pair. Additionally the subalgebra h of this pair has to be maximal isotropic or equivalently there exists an initial solution of the SC Λ 0 .

Generalized geometry
Looking closely at the exact sequence where E a denotes the vector E a i ∂ i . In conjugation with the maps we have discussed in subsection 3.1, η allows us to refine the exact sequence (3.57) to (1.2) in the introduction. All maps appearing there are nothing else than the components of a generalized frame field E A and its inverseÊ A [51][52][53]. It represents a map from E ∼ = g to the generalized tangent bundle T * M ⊕ T M . 4 We immediately read off its componentŝ . (3.60)

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The corresponding dual frame follows directly from the properties of the exact sequence and readsÊ . (3.61) Here, we denote elements of the generalized tangent bundle as V = v +ṽ with v ∈ T M and v ∈ T * M . Finally, let us write down the generalized frame and its dual in the explicit form In order to obtain the dual frame, one has to take into account σ * ω = 0 which results in  which also shows thatÊ AÎ andÊ AÎ are O(d, d) elements.
With the generalized frame and its inverse, we are able to pull the generalized Lie derivative of DFT WZW (2.10) to the generalized tangent bundle. Its elements are connected to V A by the relation Applying the dual frame on the flat derivative D A , we further obtain Plugging these two relations in the generalized Lie derivative (2.10) written in terms of flat derivatives and combines the structure coefficients of the Lie algebra g with the covariant fluxes of the generalized frame fieldÊ A . Pulled to the generalized tangent bundle, the structure coefficients FÎĴK play the role of fluxes. We use the following conventions to write down the components of the H-, f -, Q-and R-flux in a compact form. We apply the same notation to the twist, for which the fluxes are decorated by the subscript F , and to the contributions from the generalized frame field labeled by the subscriptÊ.

Deformations
In the last subsections, we constructed solutions of the SC and linked them to GG. But can the formalism presented in this paper provide any additional information which are not manifest in a GG description? One way to approach this question, is to check whether the SC solution for a given group manifold G is unique.
which gives rise to the expansion From this series, we immediately read off the g-valued two-forms by comparing with (A.11). We are only interested in those of them which do not spoil the subalgebra h. Thus only transformations with δ n F abc = 0 are allowed. Finally, we have to check whether the restricted forms fulfill the constraints for a valid deformation of h outlined in appendix A.

Examples
In this section, we present explicit examples for the techniques discussed in the last subsection. We stick to d = 3, where all Lie algebras g which admit an embedding into o(3, 3) are classified [54] and the corresponding group manifolds G are constructed [35]. Our first example is the canonical one, the torus with H-flux. It is generated by the Lie algebra cso(1, 0, 3) and gives rise to a chain of T-dual backgrounds which were studied extensively in the literature [23,42,43,55,56]. Except for the torus with R-flux, all of them are captured by deformations of the maximal isotropic subalgebra h as described in section 3.4. Furthermore, we consider the standard example of DFT WZW , the three-sphere S 3 with H-flux. 5 Here, the Lie algebra is so(4) and its subalgebra h = so(3) does not admit nontrivial deformations. However, it is still possible to find a T-dual solution of the SC after removing a discrete subgroup from the physical space M =SO(4)/SO(3). All results are in perfect agreement with the topology changes one would expect from T-duality [1]. By calculating the characteristic classes of the H-principle bundles of these examples, we also see that the generalized tangent bundle is in general twisted.

Torus with H-flux
A torus with h units of H-flux is characterized by the Lie algebra g=cso(1, 0, 3) with the non-vanishing commutator relations 6 [t a , t b ] = h abc t c . (4.1) The three generators t 1 , t 2 , t 3 are in the center of this algebra and form an abelian, maximal isotropic subalgebra h. In order to construct a SC solution for this background, we first obtain all elements of G by applying the exponential map to a faithful matrix representation of the Lie algebra [35], resulting in Note that the group CSO(1, 0, 3) we get from this procedure is not compact. In order to compactify it, we mod out the discrete subgroup CSO(1, 0, 3, Z) (acting from the left) which results in the identifications for the coordinates on the group manifold. Their derivation is explained in appendix B. Note that they are respecting the restriction (3.31) on transition functions and therefore give rise to a global SC solution once the SC is solved locally in every patch. In order to find this local solution, we now follow the steps outlined in section 3 and try to obtain a B-field such that the field strength F = F a t a for the connection is zero. This is not very complicated because the gauge group is abelian and the field strength F a = dA a vanishes for In this case A a is pure gauge and can be written in the form A a = dλ a with (4.6) 6 We use the convention 123 = 1 for the totally antisymmetric tensor. Its indices are raised and lowered with the identity δ ab /δ ab .

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After applying the transformation g → g exp(t a λ a ) to the group elements g = mh, we obtain the components of the background generalized vielbein defined in equation (3.15). In conjunction with the B-field, they are sufficient to completely fix the frame fieldÊ A . A shortcut to obtain the same result is to start from the coset representative According to our discussion is section 3.2 this choice has to solve the SC. Indeed, using it we obtain the same background generalized vielbein as in (4.7). If we calculate the twist (3.72) of the generalized Lie derivative, the only non-vanishing contribution is Thus, we obtain a H-twisted Lie derivative. This results is surprising. Why are the h units of H-flux on the torus distributed in such a strange way between the frame field and the twist? To answer this question, we take a closer look at the contribution coming from the former. Remember equation (3.43) for the B-field. We have a SC solution with A = 0 and can apply it to obtain F a is the field strength for the gauge connection A, defined in (3.32). Because the subgroup h is abelian for this background, each of the three F a represents a characteristic class of a circle bundle over M . As such they have to be elements of H 2 (M, Z) and indeed taking into account (4.7) results in Each of them gives an integer if integrated over the torus spanned by x 2 , x 3 / x 1 , x 3 or x 1 , x 2 , respectively. Hence, the H-flux contribution from the frame field has to be A non-trivial H-flux in H 3 (M, Z) describes a gerbe structure (see for example. [59] for an introduction). It would be interesting to better understand how the data of the H-principal bundle is related to this gerbe in general. From (3.72) it is obvious that the twist and the contribution from the frame fields have to sum up to the right among of H-flux, (4.14) On way to get ride of the twist contribution H F is to consider a partial solution of the SC. This breaks the T-duality group from O (3,3) to the subgroup O (2,2). For the coordinates on the group manifold G which transform in the 6 of the former, this induces the branching 6 → (2, 2) + 2 (1, 1) .
Now remove the generator t 1 from the Lie algebra and only keep commutators involving the remaining ones. Doing so, we obtain a 5 dimensional Lie algebra. Its Lie group G has the coordinate irreps (2, 2) + (1, 1) where the (1, 1) part always solves the SC. This group manifold describes the extended space (not fully doubled anymore) as doubled twotorus with coordinates x 2 , x 3 ,x 2 ,x 3 fibered over a circle with coordinate x 1 . Now, the subalgebra algebra h has only two abelian generators. This removes one of the three Chern classes from equation (4.10) and results in the right among of H-flux. At the same time, the twist vanishes. As we will see in a second this modification is in agreement with the fact that we cannot perform three independent T-duality transformations and still solve the SC. Thus, all SC solutions are indeed captured by O(2, 2). A drawback is that one has to modify the generalized tangent space from T * M ⊕ T M to an adapted version. This renders the explicit equations derived in section 3.2 and 3.3 invalid. However the general procedure outlined there still applies.
Let us now study possible deformations of this solution. To this end, we choose the coset element T A B generated by to deform the structure coefficients of g according to (3.75). For the resulting series (3.76), we find δ n F abc = 0 and only c n with even n do not vanish. Thus, we substitute λ → √ λ in (3.74) and c 2n → c n to obtain the deformation series (A.11) with the coefficients Only this second part is relevant to our discussion because SO(2, Z)⊂SO(3) rotations do not change the subalgebra h. Hence, we only keep T 12 which generates a T-duality transformations along the directions x 1 , x 2 and results in a torus with Q 12 3 flux. There are two other, independent choices of t A B in (4.16), They mediate T-dualities along the directions x 1 , x 3 and x 2 , x 3 , respectively. After applying any of them, h is equivalent to the Heisenberg algebra. Starting over again from this algebra, we do not find any other non-trivial deformations.
We are still missing tori with f -and R-flux, as they should arise from the T-duality chain where T i denotes a T-duality transformation along the coordinate x i . In order to obtain the former, we have to remember that the deformations we performed above only contain SO(3)×SO(3) group elements with determinate 1. However, from DFT we know that Tduality transformations are actually covered by O(d)×O(d) transformations. Thus, we have to add an additional element T 123 which acts as of the Lie algebra (4.1), the subalgebra h is still abelian and the corresponding connection A vanishes without switching on a B-field. For the components of the background generalized vielbein, we now obtain Since the vielbein E a i is not trivial, there exists a non-vanishing geometric flux

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We rewrite it in terms of three different two-forms f a = 1 2 f bc a E b i E c j dx i ∧ dx j and notice that the only non-vanishing contribution is given by which represents the first Chern class of a circle bundle over the base T 2 in the directions x 1 and x 2 . The twist of the generalized Lie derivative on the generalized tangent bundle vanishes and we have no H-flux contribution from the frame fieldÊ A either (as expected for the twisted torus). Hence, T-duality has exchanged the role of where H denotes the non-trivial H-flux (4.13) in the T-dual frame. This result is in perfect agreement with the literature about T-duality for non-trivially fibered circle bundles [1,23]. Our initial conjecture that different solutions of the SC lead to T-dual GGs is confirmed. It is very important to keep in mind that we do not have to choose any generalized metric, in order to perform these calculations. Because we are still on the same group manifold the two solutions (4.7) and (4.25) are related to each other by the change of coordinates All fields on the doubled space may only depend on the variables x 1 , x 2 , x 3 . Otherwise they would violate the SC. If there exist fields which only depend on x 1 , x 2 , they solve it before and after the coordinate transformations (4.29). For these fields we can freely choose a duality frame. This observation is in perfect agreement with the common fact that T-duality is only allowed along isometries of the background. After identifying Tduality transformations with coordinate changes (diffeomorphisms), it becomes clear why the covariance of DFT WZW under standard diffeomorphisms is so important: it make Tduality manifest.
Torus with Q-flux. Finally, we perform the deformation in order to obtain a torus with Q-flux. As for the twisted torus, the connection A is flat for B = 0. We repeat all the steps outlined above and obtain a trivial generalized frameÊ A . From (3.72) it follows directly that the twist F A C = F AB C is equivalent to the structure coefficient of the Lie algebra g. Its non-vanishing contribution is As expected, we obtain a background with Q-flux which can be translated into the Poisson algebra

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governing the coordinates of the physical subspace M [60][61][62][63][64]. Taking this identification into account, we conclude that we obtain a non-commutative two-torus (x 2 , x 3 ) which is fibred over a circle with the coordinate x 1 . The same conclusion was drawn in [65]. Again, this frame can be related to the torus with H-flux by the coordinate transformation Because the generalized frame field (3.62) lacks a contribution from an anti-symmetric bivector β ij , it cannot implement Q-flux. However, it would be interesting to extend the presented formalism to include it. In this paper, we start from a Lie Group with the corresponding Lie algebra and obtain a Courant algebroid. It can be rewritten in terms of two Lie algebroids fulfilling a compatibility condition. One for the tangent and one for the cotangent bundle. They give rise to a Lie bi-algebroid with the structure of a Courant algebroid (see [66] for a review). For the torus with H-and f -flux, the Lie algebroid of the tangent bundle is non-trivial while the one for the cotangent bundle is trivial. For the Q-and R-flux frames, this situation is flipped. Thus, our present formalism seems to be unable to produce non-trivial Lie algebroids for the cotangent bundle. This is not surprising because we implemented the Lie algebra g on the tangent bundle of the group manifold only. A generalization would consider the dual Lie algebra g * , too. Combining it with g one could obtain a Lie bi-algebra corresponding to a Poisson-Lie group. Starting from this generalized structure instead of a Lie group only, it could be possible to incorporate Qand R-fluxes on the same footing as H-and f -flux. However, this modification is beyond the scope of this paper.

Three-sphere with H-flux
To describe a S 3 with h units of H-flux, we consider the Lie algebra g = so(4) defined by the non-vanishing commutators We identify the maximal isotropic subalgebra h = so(3) and assign the remaining generators t a to m. These two subsets form a symmetric pair. Hence, the technique to find a flat connection from section 3.2 applies to this example. For the following discussion, it is convenient to remember the isomorphism SO(4) ∼ = SU(2) × SU(2)/Z 2 . In order to implement it, take a point in R 4 with coordinates y 0 , . . . , y 3 and write it in terms of the 2×2 matrix Y = 1y 0 − iσ j y j (4.36)

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with σ j , j=1,2,3 , denoting the Pauli-matrices. An arbitrary SO(4) action on this point can be rewritten as Because the SU(2)×SU(2) elements (g L , g R ) and (−g L , −g R ) mediate the same transformation, one has to mod out Z 2 in (4.35). Furthermore, assume that g R and g L are generated by They reproduce the algebra used in appendix A of [33] to discuss DFT WZW for the SU(2) WZW-model. This representation is well suited make contact with the underlying CFT description of the target space. By combining t a = t L a + t R a and t a = t L a − t R a , (4.40) we recover the non-vanishing commutators in (4.34).
We choose a coset representative m of the form (3.45). As shown in section 3.2, it automatically solves of the SC. Applying the isomorphism (4.35), we write it as How is m related to a three-sphere? From an abstract point of view, the isomorphism g m SU(2) ∼ = S 3 is well known. Here, we want to make this connection manifest and derive a map from points on the three-sphere to coset representatives m. To this end, we first expand g m = 1x 0 − iσ j x j (4.42) in the same way as Y . Note that the additional constraint (x 0 ) 2 + (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 = 1 holds, because g m ∈SU (2). Applying m to the unit vector represented by Y = 1 gives rise to Y = g m g m or equivalently y 0 = 2(x 0 ) 2 − 1 and y i = 2x 0 x i . (4.43) If we choose hyperspherical coordinates y 0 = cos φ 1 y 2 = sin φ 1 sin φ 2 cos φ 3 for a unit S 3 embedded in R 4 , we find the corresponding coset representative (4.41) with by inverting the relations (4.43).

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Starting from the explicit parameterization (4.46) of the coset representative (4.41), we calculate the components and of the frame field (3.62). Its B-field gives rise to a non-vanishing H-flux between the frame field and the twist of the generalized Lie derivative. Note that this result is different from the one for the torus with H-flux in (4.14). We can explain it by taking a closer look at H=SO(3)-principal bundles over three-spheres. First, split the S 3 into two overlapping patches U N /U S covering its north/south pole. On each one there exists a local section σ N /σ S . Both are patched together along a two-sphere S 2 = U N ∩ U S on the equator. Topologically distinct transition functions are classified by the second homotopy group π 2 (SO(3)) which is trivial. Thus, the bundle always admits a global section and is trivial. At the same time, the resulting generalized tangent bundle T * M ⊕ T M is not twisted and all non-trivial H-flux has to be implemented as a twist of the generalized Lie derivative.

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Let us finally discuss an alternative SC solution on G. The subalgebra h = so(3) is simple and does not admit any deformations. So we cannot apply the technique outlined in section 3.4. Still, we can modify the physical subspace M which arises as a solution of the SC to reproduce the results about T-duality on the S 3 with H-flux [1]. To this end, consider the double coset where Γ is a freely acting, discrete subgroup of SO (4). Only discrete subgroups are allowed because they do not change the dimension of M . Here, we choose Γ = Z n . Its elements k l , l = 0, . . . , n − 1, are embedded by into SO(4) and have the action z 1 → e −2πil/n z 1 and z 2 → e −2πil/n z 2 with z 1 = y 0 + iy 3 z 2 = y 2 + iy 1 with 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π and 0 ≤ ξ ≤ 4π . In these coordinates the identification above only affects ξ and we have to simply restrict its range 0 ≤ ξ ≤ 4π n (4.60) to obtain the lens space L(n, 1). Single-valued functions f (θ, φ, ξ) on the S 3 are in general multi-valued on L(n, 1). However, we can expand them in a set of functions As expected, Λ and therewith its absolute value squared are not single valued on L(n, 1). In order to get a simple expression for it in the w = 0 sector, we further assume that n = h.
In this case, we only have to evaluate the zero mode contribution (4.66) It gives rise to with the cohomology class Hence, we reproduce the topology change induces by T-duality along the Hopf-fiber of a S 3 with h units of H-flux. In agreement with [1], the dual space is a lens space L(h, 1) with one unit of H-flux. An extensive discussion of the doubled geometry of WZW-models is given in [44]. As here, the doubled space is treated as a group manifold and a projection π : G → G/H is used to obtain the physical subspace. Still, we cannot reproduce the mechanism for abelian T-duality outlined there. Our results suggest that T-duality does not simply exchange a winding and a momentum circle over a two-sphere base. Its action is more subtle due to the finite number of winding modes.

Outlook
In this paper we show how the SC of DFT WZW is solved and how its solutions are related to T-duality. Doing so, only the topology of the group manifold G enters the discussion. We do not need to fix a generalized metric and still reproduce topology changes induces by T-duality [1,23]. There are still various open questions: • The examples we present do not include backgrounds which are not T-dual to geometric one. These have geometric and non-geometric fluxes with common directions. R-flux presents an obstruction to find a maximal isotropic subgroup [27,52], but there are also examples with Q-flux like the double elliptic setup in [67] which can be treated with the techniques introduces here. It would be very interesting to study whether such backgrounds admit SC solutions or not.
• Solitonic objects like branes and monopoles are essential in studying string theory. They are characterized by topological charges which change under T-duality. An illustrative example is the NS5-brane. Take h of them sitting at a point in the transverse space R 3 × S 1 and a two-sphere S 2 which encompasses the stack in R 3 . The number of branes follows directly from the H-flux because each NS5-brane contributes one unit of H-flux. After applying T-duality along the S 1 , it becomes non-trivial fibered over the S 2 and obtains a non-vanishing first Chern class [c 1 ] = h. The resulting space is formed by h Kaluza-Klein monopoles [1]. In the same vein, the example S 3 with H-flux in section 4.2 is equivalent to a configuration with one KK-monopole and h NS5-branes. Studying these objects and also their non-geometric counter parts [68,69] is a prominent application of DFT [70][71][72]. Currently, only local solutions in terms of an explicit generalized metric are know [71]. However, there is also a very interesting global perspective as the NS5-brane/KK-monopole example above shows. We hope that DFT WZW combined with the techniques present in this paper facilitate further progress in this direction.
• By solving the SC, we obtain exact Courant algebroids. There is a lot to say about them and their connection to DFT [73,74]. Here, we only want to highlight that each Courant algebroid is linked to a Courant σ-model which plays an important role in understanding the non-associative spacetime in R-flux backgrounds [60,[75][76][77][78][79]. On the other hand a S 3 with H-flux is automatically non-associative for a finite among of flux. To see this, remember that the corresponding σ-model gives rise a SU(2)×SU(2) Kač-Moody algebra at level h. Primary fields are translated to a set of orthogonal functions on the physical space M = S 3 . Their angular momentum cannot exceed the level. After multiplying two of them, one can in general exceed the maximally allowed angular momentum. Therefore a projector is required. It renders the algebra of functions on M non-associative [80]. For h → ∞ this effect vanishes. It is directly related to α corrections because in DFT WZW we have an expansion in JHEP04(2018)128 1/h instead of α [33]. Now the speculative part of the story: T-duality along all directions of the target space should inverts the worldsheet coupling 1/h → h. As a result there is a transition from a large, semiclassical S 3 to a very small, strongly non-associative background. We have to face two major challenges in trying to test this idea. First, α corrections are not worked out in DFT WZW . Furthermore, we are not able to find solutions to the SC with non-trivial R-flux. We hope that in the future enough progress is made to over come these challenges. Additional hints in the right direction are given by [75,76,79].

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.