Generalized Metric Formulation of Double Field Theory on Group Manifolds

We rewrite the recently derived cubic action of Double Field Theory on group manifolds [arXiv:1410.6374] in terms of a generalized metric and extrapolate it to all orders in the fields. For the resulting action, we derive the field equations and state them in terms of a generalized curvature scalar and a generalized Ricci tensor. Compared to the generalized metric formulation of DFT derived from tori, all these quantities receive additional contributions related to the non-trivial background. It is shown that the action is invariant under its generalized diffeomorphisms and 2D-diffeomorphisms. Imposing additional constraints relating the background and fluctuations around it, the precise relation between the proposed generalized metric formulation of DFT${}_\mathrm{WZW}$ and of original DFT from tori is clarified. Furthermore we show how to relate DFT${}_\mathrm{WZW}$ of the WZW background with the flux formulation of original DFT.


Introduction
For several years, dualities have became a well established instrument to study fundamental aspects of string theory and the corresponding low energy effective field theories. Hence, it is not a surprise that there is a growing interest in a theory called Double Field Theory (DFT) [2][3][4][5][6][7][8][9] which makes abelian T-duality a manifest symmetry in the low energy description of closed string theory. To this end, it seizes the idea [2,[10][11][12][13] to double the coordinates of the target space. Adding D additional dual coordinates allows to take winding excitations of the closed string on a compact background into account. Exchanging winding and momentum excitations is the mechanism underpinning T-duality on a torus and thus the doubled target space of DFT permits to capture this mechanism through a global O(D, D) symmetry. The doubling of the coordinates can be also viewed as introducing D left-moving and D right-moving closed string coordinates, where the ordinary and dual coordinates are just the sums and the differences of left-and right-moving coordinates.
However, there are still conceptual questions about the current status of DFT. They are mainly triggered by the strong constraint which is required for a consistent low energy formu lation. The strong constraint is a consequence of the toroidal background used in the original derivation [3] and it states that winding and momentum excitations in the same direction are not allowed. Violating the strong constraint, it is impossible to choose a torus radius in the corresponding direction to make all fields much lighter than the string scale. Either momentum or winding modes are heavier than the first massive string excitations and spoil a consistent truncation. On the other hand, applying the strong constraint identifies DFT with the well studied NS/NS sector of SUGRA. Thus, except for an effective rewriting, it does not give any new physical insights. Moreover, such a rewriting is also available in terms of Hitchin's gen eralized complex geometry [14,15] which is an appropriate replacement for DFT in this case. The situation is more intriguing, but unfortunately also more speculative, if one weakens the strong constraint. In this case so called non-geometric backgrounds [12,13,16,17] arise. They are partly inspired by generalized Scherk-Schwarz compactifications which give rise to gauged supergravities not accessible by flux compactifications from the SUGRA regime [11,[18][19][20][21]. Some of these backgrounds have an uplift to string theory in terms of left-right asymmetric orbifold constructions [11,[21][22][23], but in general their fate is unknown.
In order to improve this situation three of the authors proposed an alternative theory with a doubled coordinate space called DFT WZW [1]. It originates from tree-level Closed String Field Theory (CSFT) calculations up to cubic order in the fields and leading order of α on a group manifold. This theory is governed by a Wess-Zumino-Witten model on the worldsheet. In DFT WZW the doubling of the coordinates basically refers to the left-and right-moving currents of the WZW model on a group manifold. Interestingly, it turned out that this theory does not reproduce all results known from original DFT: The strong constraint, the gauge transformations and the action receive corrections from the non-trivial string background. Furthermore, the closure of the gauge algebra only requires the strong constraint for fluctuations, whereas the weaker closure constraint is sufficient for the background fields. In this way, one can obtain a consistent tree-level description of non-geometric backgrounds. All these properties suggest that DFT WZW should be considered as a generalization of original DFT. However, a direct comparison between the two at cubic level seems to be impossible. Therefore, in this paper we derive the full generalized metric formulation of the theory. Let us summarize our results in the following.
The resulting action to all orders in the fields reads where d denotes the generalized dilaton and R represents the generalized curvature scalar of DFT WZW . It incorporates the generalized metric H IJ combining fluctuations and background, the covariant derivative and the background metric H IJ with the associated structure coefficients F IJK of the group man ifold. Thus we have H IJ = H IJ +∆H IJ and the connection appearing in the covariant derivative is determined entirely by the background. In this sense, the theory presented here is manifestly background dependent. As we will discuss in section 5, this is not a contradiction in being a generalization of DFT which is background independent once the strong constraint is invoked. We show that the proposed action (1.1) is invariant under the generalized diffeomorphisms where L ξ denotes the generalized Lie derivative of the theory. In all calculations, we assume the strong constraint to be fulfilled for the generalized dilaton d, the generalized metric H AB , the parameter ξ A of the generalized Lie derivative and arbitrary products of them. The strong constraint only applies to quantities in flat indices. To switch between curved and flat indices the generalized vielbein E A I of the background is used. Additionally, we also apply the Jacobi identity for the structure coefficients of the background. Besides generalized diffeomorphisms, (1.1) is manifestly invariant under 2D-diffeomorphisms with L ξ denoting the ordinary Lie derivative. In view of this, DFT WZW seems to implement a non-trivial extension of the DFT gauge algebra as proposed by Cederwall [24,25]. Still, there exists an important difference. Whereas Cederwall considered only torsionless covariant deriva tives, the covariant derivative (1.3) exhibits a torsionful connection. One of the objectives of this paper is to clarify the relation between background dependent DFT WZW and original DFT. We will succeed to identify DFT WZW with the generalized met ric formulation of DFT [26] under two special assumptions: First, a distinguished generalized vielbein which fulfills the strong constraint of DFT is required and second an extended strong constraint linking background fields b and fluctuations f , has to be imposed. It is important to note that this constraint is totally optional in the framework of our theory. Hence, it is reasonable to suspect that there exist valid field configurations in DFT WZW that go beyond DFT. This statement even holds, if the background group manifold is purely geometric or T-dual to a geometric one. Identifying the two theories under the assumptions mentioned above, we confirm the background independence of DFT suggested in [5]. This background independence is a result of the very restrictive strong constraint in DFT which renders it equivalent to SUGRA. The organization of this paper follows the outline given in the last paragraph. After a short review of the DFT WZW cubic action and the required notation, section 2 presents the generalized metric formulation of the action and its gauge transformations. Section 3 discusses the equations of motion of this action. Further, it derives the generalized curvature scalar and the generalized Ricci tensor. In section 4, we proof the invariance of the action under generalized diffeomorphisms and 2D-diffeomorphisms. At last, we show the equivalence of our theory and original DFT in section 5. A small outlook, discussing the potential and possible applications of DFT WZW concludes the paper in section 6.

Generalized metric formulation
Starting from the results derived in [1], we derive the generalized metric formulation of the DFT WZW action in this section. As a preliminary, subsection 2.1 reviews the most important aspects of the cubic action derived in [1] and introduces the required notation. Although already discussed in [1], we shortly present the gauge transformations and the C-bracket governing the gauge algebra in subsection 2.2 before discussing the new results for action in subsection 2.3.

Review of cubic action and notation
The cubic action and gauge transformations were derived at the leading order of α from CSFT in [1]. The starting point are fields ab that can be considered as fluctuations around the WZW background. The indices a andb refer to the adjoint representation of the corresponding group G L × G R . In addition we also introduce gauge parameters λ a and λā. In contrast to the toroidal case, one does not consider momentum and winding modes but one considers different representations R = (r L , r R ) of G L × G R . Here, we do not use the form stated in [1], but instead perform the field redefinition ab → −2 ab , λ a → 2λ a and λā → 2λā  Besides, further rescaling in the definitions given later in this subsection, this field redefinition helps to get rid of a 1/2 factor which arises in [1] between the DFT and the DFT WZW results.
To allow a clear distinction between background fields and fluctuations, we have changed the notation of [1]. Now,d denotes fluctuations of the generalized dilaton d =d +d which combines the background fieldd and the fluctuations. As a consequence of level-matching in closed string theory, the fields ab ,d and the gauge parameters λ a and λā have to fulfill the strong constraint where · not only denotes the mentioned field but also arbitrary products of them. On the world sheet, the theory is governed by a CFT with two independent, a chiral (left mover) and an anti-chiral (right mover), Kač-Moody current algebras. The structure coefficients of their central extensions g L × g R are denoted by F ab c and Fābc. Bared and unbared indices allow to distinguish between the algebras for the left and right moving part of the closed string. These indices run from 1 . . . D, the dimension of the group manifold used as background. The integration in (2.2) is performed over a product manifold combining the Lie groups G L × G R associated to the Lie algebras g L × g R . This manifold is parameterized by the 2D coordinates X I = (x i xī) and is equipped with the metric S AB = 2 η ab 0 0 ηāb and its inverse S AB = 1 2 in flat indices. It combines the killing metrics η ab / ηāb of the Lie algebras g L / g R which are used to raise flat indices. Moreover, it is very convenient to introduce the vielbein Finally H IJ , whose determinante H is used in (2.2), is defined as the curved version of S AB . As a consequence of the rescaled flat metric S AB , H IJ differs by a factor 2 from the definition in [1]. To keep the action integral (2.2) invariant, one has to perform the compensating change of variables X I → X I / √ 2. Besides the background metric S AB in flat indices, it is convenient to introduce the metric η AB = 2 η ab 0 0 −ηāb and its inverse η AB = 1 2 to raise and lower doubled indices. In combination with the doubled flat derivative (2.8) it e.g. allows to express the strong constraint (2.5) in the compact form

Gauge transformations
Switching from the notation with bared and unbared indices to doubled indices, generally sim plifies the equations in DFT WZW a lot. In this respect, the strong constraint (2.11) is a toy example. More drastic is the effect on the gauge transformations (2.4). In order to express them in doubled notation, we follow [1] and introduce the symmetric, O(D, D) matrix called generalized metric. It is generated by which embeds the fluctuations ab into a tensor with doubled indices. Furthermore, we define the flat covariant derivatives where combines the structure coefficients defining the Kač-Moody algebras for the strings left and right moving parts. At this point, let us recall the conventions from [1]: D A , F AB C and ξ A are considered as "fundamental" objects, meaning their bared and unbared components do not receive additional minus signs or prefactors. From these quantities all others are derived by raising/lowering the doubled indices with the η-metric. A simple example is Now, we expand the generalized metric (2.12) into components up to cubic order in the fields. Plugging this expansion into one recovers the gauge transformations (2.4) up to additional terms which are not linear in the field or the gauge parameter. The same holds for the C-bracket

Action
In this subsection, we rewrite the action (2.2) in terms of the generalized metric. The guiding principle is inspired by the results for the gauge transformations and the C-bracket discussed in the last subsection: in the expressions known from traditional DFT, one has to substitute partial derivatives by covariant derivatives (2.14). Taking into account the original DFT action in the generalized metric formulation [26] and following this principle, the action should read Subsequently, we proof that, up to cubic terms in the fields, this action indeed reproduces (2.2) up to a missing term that has to be added to (2.20). To keep this straightforward though cumbersome calculation as traceable as possible, we begin with terms containing two flat deriva tives like e.g.
We further simplify the calculation by first considering the term which gives rise to after plugging in the components of S AB and H AB , according to (2.6) and (2.17). From the first to the second line in (2.23), we perform integration by parts by applying the rule (2.24) It automatically arises, if one splits the generalized dilaton into the background partd and the fluctuationsd around this background. Performing integra tions by parts again and dropping the terms in quartic order in the fields, we obtain Now, it is straightforward to read off the remaining terms of (2.21), namely Here and in the following, O(. . . ) is suppressed for brevity. The last term in (2.26) cancels against a term arising in the expansion of Next, we turn to the term for which the calculations are more cumbersome. Using the commutation relations for flat derivatives and performing integration by parts, we finally obtain the result All remaining terms in the action (2.20) contain covariant derivatives acting on the generalized dilaton d. Its background partd is covariantly constant and the fluctuationsd transform like a scalar. Thus, we are able to identify In combination with the expansion (2.17) of H AB , this identity gives rise to Taking into account the prefactor e −2d , we obtain where we applied the relation √ The last term in (2.20), which contains two flat derivatives, gives rise to It indeed matches with the action (2.2) after dropping all terms depending on the structure coefficients F abc and Fābc.
Let us now consider these terms so that we have to consider the full covariant derivative instead of only using its flat derivative part. Let us start with where the second term in the first line vanishes due the total antisymmetry of F ABC and the symmetry of H AB . The third term is zero due to the unimodularity condition which the structure coefficients have to fulfill [1]. At this point, we come to the more challenging part In the subsequent computation, we ignore all terms which contain more than one flat derivative, because we already discussed these contributions above. The first part of (2.40) gives rise to where the second term on the right hand side is equivalent to after using the symmetry of H AB and relabeling the indices. For the fourth term, we use the total antisymmetry of the structure coefficients to yield Applying these two substitutions, (2.41) simplifies to For the second part of (2.41), we obtain in a similar fashion After combining these results, we finally get The first line on the right hand side exactly reproduces the structure coefficients dependent terms in the cubic action (2.2), but the second line has to be canceled to successfully reproduce the action. Achieving this is done by adding the term to the naive action (2.20). Because it neither affects the equations of motion nor the gauge transformations, we do not care about the constant shift. If desired, it can be canceled by an additional term in the action. We close this section with the complete action of DFT WZW

Equations of motion
After deriving the full action of DFT WZW in the last section, we now discuss its equations of motion. It is convenient to split them into two independent parts. First, we present the variation of the action (2.48) with respect to the generalized dilaton d in subsection 3.1. It gives rise to the generalized curvature scalar R. Furthermore, we show how the action can be rewritten in terms of this scalar. In the second step, we perform the variation with respect to the generalized metric H AB in subsection 3.2. Just as in the generalized metric formulation of DFT [26], we have to apply an appropriate projection, taking into account the O(D, D) property of the generalized metric, to obtain the generalized Ricci tensor R IJ .

Generalized curvature scalar
Following [26], we define the generalized scalar curvature R of DFT WZW using the variation of the action (2.48) with respect to the generalized dilaton d. A straightforward calculation gives rise to In order to proof the invariance of the action (2.48) under generalized diffeomorphisms in the next section, it is very convenient to express it in the form To this end, we rewrite (2.48) as where the last term is a vanishing boundary term. Due to the compatibility of the covariant derivative with the generalized vielbein, it is trivial to express the generalized scalar curvature in curved instead of flat indices. One only has to relabel the indices to obtain the desired result (1.2) stated in the introduction. The generalized dilaton part of the equation of motion reads

Generalized Ricci tensor
Now, we consider the variation of the action (2.48) with respect to the generalized metric H AB . In analogy to (3.1), we consider As discussed in [26], the variation δH AB is symmetric and thus it is sufficient to study the symmetric part of K AB only. Performing the variation explicitly and afterwards symmetrizing K AB gives rise to Furthermore, the O(D, D) constraint has to be preserved under the variation [26]. This implies that only a certain projection of K AB gives rise to the equations of motion. Hence, it is necessary to introduce the projection operators which are used to define the generalized Ricci tensor This projection cancels the term in the last line of (3.7). Thus, we find a generalized Ricci tensor whose structure matches the one of toroidal DFT. However, all partial derivatives have to be replaced with covariant ones.

Local symmetries
The CSFT derivation of DFT WZW in [1] was very challenging. The recasting of the action and the gauge transformations in section 2 is a good, first indication that everything is consistent: All the different terms with bared and unbared indices integrate nicely into doubled objects. However, a much more important consistency check is the invariance of the action (2.48) under the gauge transformations (2.18). If all previous calculations were performed correctly, the CSFT framework guarantees this invariance up to cubic order in the fields. As we will show in subsection 4.1, it even holds to all higher orders introduced by the generalized metric formulation. Besides generalized diffeomorphism invariance, the action is also manifestly invariant under 2D-diffeomorphisms, as we proof in subsection 4.2.

Generalized diffeomorphisms
It does not matter whether one proofs the invariance under gauge transformations for the action (2.48) or (3.3). Both only differ by a vanishing total derivative. We choose the latter one, with the generalized curvature scalar R. Proving its invariance, requires two step: First, we show that R transforms as a scalar under generalized diffeomorphisms. Second, we consider the remaining term e −2d and show that it transforms as a weight +1 scalar density. In order to show that the generalized curvature (3.2) is a scalar under generalized diffeomor phisms, we have to compare its transformation behavior under gauge transformations with the results we expect from generalized diffeomorphisms mediated by the generalized Lie derivative. The failure of a quantity V to transform covariantly under generalized diffeomorphisms reads where L ξ is the generalized Lie derivative hold. Furthermore, ∆ ξ is linear and fulfills the product rule Please note that the gauge transformations δ ξ act on the fields H AB andd only, whereas the generalized Lie derivative L ξ acts on the full tensorial structure. As an instructive example take e.g.
We now calculate ∆ ξ for all sub-terms appearing in the generalized curvature scalar (3.2). Finally, we combine these results, using the product rule and the linearity of ∆ ξ to compute ∆ ξ R. We begin with and since holds, we only need to consider On the right hand side, we canceled all terms of the form They vanish due to the strong constraint (2.11). Combining these results, we are finally able to calculate ∆ ξ of the naive generalized Ricci scalar (3.2) without the F ACE F BDF H AB S CD S EF term. It is denoted asR and its failure to transform as a scalar under generalized diffeomorphisms reads Here, we ordered the terms according to the number of derivatives. All terms with three flat derivatives vanish in the same way as they do for toroidal DFT [26]. The third line of (4.12) vanishes due to the Jacobi identity Additionally, one is able to rewrite the first line as showing that it is zero due to the Jacobi identity, too. Simplifying the remaining terms in which gives rise to Further, due to the antisymmetry of the structure coefficients we identify and obtain The last term vanishes due to 4.11. For the remaining two terms we use the identity which yields This non-vanishing failure ofR to transform like a scalar should be canceled by the term that we have not taken into account yet. Indeed, ∆ ξ applied on this term gives rise to after remembering δ ξ S AB = 0 (gauge transformations act on fluctuations only, but not on background fields [1]). Ultimately, we obtain the desired result which proofs that the generalized curvature scalar 3.2 is indeed a scalar under generalized dif feomorphisms.
In addition to R, we have to check the transformation behavior of the factor e −2d in the action (3.3). To this end, we first rewrite the generalized Lie derivative of the dilaton fluctuations (2.18) in terms of covariant derivatives where the last term vanishes due to the unimodularity of the structure coefficients. Next, we consider where we take into account that the background fieldd is not affected by gauge transformations. With L ξd written in terms of covariant derivatives, it is trivial to switch to curved indices. Doing so and plugging in (4.24), δ ξ e −2d reads as explained in [1]. Thus, we see that e −2d transforms like a scalar density with the weight +1 and the integral over the product e −2d R, which is equivalent to the action, is invariant. Besides the action, the generalized Lie derivative (4.2) transforms covariantly under general ized diffeomorphisms. Indirectly, this property has already been proven by showing the closure of the gauge algebra in [1]. However to make it more explicit, we consider In combination with (4.28) it vanishes after rewriting the C-bracket in terms of the generalized Lie derivative and the trivial gauge transformation −1/2∇ I (ξ J λ J ).

2D-diffeomorphisms
Besides the generalized diffeomorphisms discussed in the previous subsection, one can change the coordinates parameterizing the fields of DFT WZW through the standard Lie derivative. This gives rise to a 2D-diffeomorphisms under which the action (3.3) is even manifestly invariant. In order to proof this claim, we follow very similar steps as in the subsection 4.1. However, in this case we will not apply the strong constraint in any of the following steps. Again, we start by introducing the failure of an arbitrary quantity V to transform covariantly. Here, we use the standard Lie derivative L ξ instead of the generalized Lie derivative. The transformation behavior of the generalized vielbein E A I and the generalized dilaton fluctuationsd is given by From these two equations, we see that E A I transforms as a vector andd as a scalar under 2D-diffeomorphisms. Next, we check the failure to transform as a covariant quantity. Being called a 'covariant' derivative, this failure should vanish of course. We start by calculating the first term in (4.35) and obtain The second terms is a bit more challenging. In order to evaluate it, we need the definition of the Christoffel symbols where Ω IJK denotes the coefficients of anholonomy in curved indices. With these definitions at hand, one obtains Thus, (4.35) gives rise to the expected result and ∇ I is indeed the covariant derivative under 2D-diffeomorphisms. Even though we have shown the vanishing ∆ ξ of a covariant derivative applied on a vector, this result generalizes to arbitrary tensors. Especially, the failures vanish. The last ingredient in the definition of the generalized curvature scalar (3.2) are the structure coefficients F IJK . Fortunately, their failure to transform covariantly vanishes, too. Applying the linearity and the product rule of ∆ ξ , we immediately obtain which proofs that the product e −2d R transforms as a scalar under 2D-diffeomorphisms. For the action (3.3) to be invariant, the remaining factor e −2d has to transform as a weight +1 scalar density. Indeed, we have e −2d = |H| , (4.45) which exactly transforms in the right way. Hence, the DFT WZW action exhibits a manifest 2D-diffeomorphism invariance.
Containing covariant derivatives only, the generalized Lie derivative (4.2) transforms covari antly, too. Hence, it fulfills Rewriting this equation, we obtain giving rise to the algebra which links 2D-diffeomorphisms and generalized diffeomorphisms. Equipped with this algebra, our theory implements an extension of the DFT gauge algebra proposed by Cederwall [24,25]. However, there are some important differences we would like to comment on. Cederwall considered a covariant derivative without torsion on an arbitrary pseudo Riemannian manifold in order to define a generalized Lie derivative formally matching the one of DFT WZW . Applying the Bianchi identity without torsion he shows in full generality that the gauge algebra closes. We consider a torsionful covariant derivative on a group manifold, a very special case of a pseudo Riemannian manifold. Interest ingly, the Bianchi identity with torsion reproduces on the group manifold the Jacobi identity which we used to show the closure of the DFT WZW gauge algebra and the invariance of the action under generalized diffeomorphisms. Thus, one is inclined to conjecture that the whole formalism presented here is not limited to a group manifold as background but could hold for arbitrary pseudo Riemannian manifolds.

Transition to original DFT
Assuming the geometric group manifold as a background, in this section we study the connection between DFT WZW and the original formulation. A link between them was already conjectured in [1], but no explicit calculation was provided yet. Now, with the generalized metric formulation available, we prove that under an additional constraint both theories can be identified. For that purpose, first we introduce a distinguished generalized vielbein in subsection 5.1. Afterwards, we discuss an additional constraint that links the background fields with the fluctuations around it. We call it the extended strong constraint. As subsection 5.2 shows, this constraint allows us to identify the covariant fluxes F ABC of the DFT flux formulation [7,21,27] with the structure coefficients F ABC of the group manifold. Applying the extended strong constraint, in subsection 5.3 we prove the equivalence of the gauge transformations and the action in both theories. In this context, we will briefly discuss the background independence of DFT.

Appropriate generalized vielbein
The starting point for the following discussion is a background generalized vielbein E A I fulfilling the strong constraint of DFT. Due to 2D-diffeomorphism invariance proven in section 4.2, one is not forced to parameterize it with the left/right moving coordinates x i /xī. Instead, we choose the momentum x i and windingx i coordinates which are common in the generalized metric formulation of DFT [26]. They give rise to A canonical choice for the vielbein in the DFT flux formulation [7,21,27] is The strong constraint of DFT requires that it only depends on half of the coordinates. Without any loss of generality, we choose EÂ I to depend on the momentum coordinates x i . Note that a hat over a doubled index indicates that the η-metric are used to lower and raise this index. In order to identify this representation of η with the diagonal form (2.10) common in DFT WZW , we apply the coordinate independent O(2D) rotation It leaves the background metric invariant and thus yields Switching to curved indices, SÂB gives rise to the generalized metric It is important to note that the canonical generalized vielbein (2.7) of DFT WZW is not an O(D, D) element, because it gives rise to different representations of the η-metric in flat and curved indices, namely This is an apparent problem, if one tries to compare DFT WZW and DFT. A short calculation shows that the generalized vielbein defined in (5.2) fixes this problem. It fulfills the relation The remaining independent components F ab c and F abc vanish. Next, we switch from FÂBĈ to F ABC by applying the transformation M AB defined in (5.4). Doing so gives rise to In the second row of E A I , we drop the bar over the index a of e ai and e a i respectively to emphasis that, in contrast to (2.7), we use the left mover vielbein only. It is connected to the one for the right movers by the O(D) transformation where K denotes the Killing form Tr(ad x ad y ) 2h ∨ , with x, y ∈ g (5. 16) introduced in [1] and g is the group element parameterized by the coordinates x i . This transfor mation is embedded into which 'recovers' the correct index structure. Due to the coordinate dependence of this transfor mation, it modifies the coefficients of anholonomy according tõ After some algebra and keeping the definition t a = −tā in mind, we obtain , t a )e a i = e a i F abc (5.19) and finally

Extended strong constraint
There is still a small but peculiar difference in the two definitions of the structure coefficients In order to identify them even so, first note that Ω ABC is antisymmetric with respect to its last two indices due to O(D, D) property (5.8). Thus, we are able to write Moreover, the purpose of F ABC in DFT WZW is to define the commutator relation between flat derivatives. Thus, it is sufficient to study where · denotes arbitrary products of fluctuations AB ,d and the gauge parameter ξ A , which we also consider as a fluctuation. In DFT WZW , the strong constraint only acts on these fluctuations, whereas it does not apply for the background or the relation between background and fluctu ations. However, we can of course introduce an additional constraint, the so called extended strong constraint linking background fields b with fluctuations f . It restricts all valid field configurations in DFT WZW to a particular subset which allows to cancel the last term in (5.25) and therefore to identify F ABC = F ABC . Furthermore, it allows to cancel the last term in the strong constraint in curved indices giving rise to which is apparently equivalent to the strong constraint in the original DFT formulation.

Gauge transformations and action
Using the covariant fluxes F ABC instead of the structure coefficients F ABC , we have to recalculate the Christoffel symbols of the covariant derivative. To this end, we solve the frame compatibility condition which gives rise to For this connection, the generalized torsion vanishes. The latter links the C-bracket of DFT WZW and DFT. Thus, both theories share besides the strong constraint (5.27) the same gauge algebra, too. This also holds for the generalized Lie derivative, which can be derived from the C-bracket as Even if the Christoffel symbols Γ IJ K get modified, they still keep their transformation behavior requiring ∂ I ξ J ∂ I f = 0 and ∂ I ξ J ∂ I E A K = 0 or ∂ I ξ K = const. . By default this is the case, since the backgrounds in DFT WZW arise from a generalized Scherk-Schwarz ansatz which posses (5.40) as a consistency constraint. Subsequently, we show that the action S of DFT WZW in curved indices can be written as S = S DFT + S ∆ with the traditional DFT action Here, we also applied (5.40) to get rid of derivatives acting ond. After these substitutions, ∆ reads ∆ = H IJ Ω IKL Ω KL J + Ω IJ K Ω L LK + ∂ K Ω IJ K + ∂ I Ω K KJ . We will now calculate K IJ . To this end, it is convenient to switch from curved to flat indices. In flat indices, the strict left/right separation of the covariant fluxes (5.21) allows us to identify which immediately gives rise to