Dual Double Field Theory

We present the dual formulation of double field theory at the linearized level. This is a classically equivalent theory describing the duals of the dilaton, the Kalb-Ramond field and the graviton in a T-duality or O(D,D) covariant way. In agreement with previous proposals, the resulting theory encodes fields in mixed Young-tableau representations, combining them into an antisymmetric 4-tensor under O(D,D). In contrast to previous proposals, the theory also requires an antisymmetric 2-tensor and a singlet, which are not all pure gauge. The need for these additional fields is analogous to a similar phenomenon for"exotic"dualizations, and we clarify this by comparing with the dualizations of the component fields. We close with some speculative remarks on the significance of these observations for the full non-linear theory yet to be constructed.


Introduction
The duality transformations relating a field strength to its Hodge-dual, interchanging Bianchi identities and field equations, are ubiquitous in gauge theory, supergravity and string theory. For instance, the electromagnetic duality in four dimensions is essential for the S-duality of N = 4 super-Yang-Mills theories. Moreover, in order to define the worldvolume dynamics of certain branes, it is necessary to replace some of the standard p-form gauge potentials of string theory by their duals.
For a p-form potential, this dualization is straightforward: one simply replaces its (p + 1)-form field strength by the Hodge-dual of the field strength of the dual (D − p − 2)form. For instance, the Kalb-Ramond 2-form B 2 of closed string theory in D = 10 can be dualized to a 6-form B 6 , which in turn couples to the NS5-brane. Therefore, from the point of view of the full (non-perturbative) string or M-theory neither the 2-nor 6-form is more fundamental, suggesting that a democratic formulation in which they appear on equal footing is more appropriate.
Remarkably, taking into account further dualities or symmetries of string theory, such as T-duality, then implies that even more fields of a more exotic nature are needed. For instance, under the T-duality group O(D, D) the 2-form B 2 transforms into the metric. In fact, in double field theory, which makes the O(D, D) symmetry manifest, the metric and 2-form are part of an irreducible object, a generalized metric or generalized frame. Thus, when dualizing B 2 into B 6 , O(D, D) covariance requires that we also dualize the graviton into a 'dual graviton'. While at the linearized level there is a straightforward procedure to dualize the graviton [1,2], leading to a field living in the mixed-Young tableau representation (D − 3, 1), 1 there are strong no-go theorems implying that at the non-linear level some new ingredients are needed [3,4].
In this paper we perform the dualization of double field theory (DFT) [5,6,7] (see [8] for reviews) at the linearized level, thereby capturing in particular the dual potential B 6 and the dual graviton in a T-duality covariant way. While we restrict ourselves to the free, quadratic theory, we believe that our results give important pointers for the full non-linear theory. The construction of the non-linear theory would be necessary in order to describe, for instance, the world-volume dynamics of Kaluza-Klein monopoles (in a way that is compatible with T-duality), for which the dual graviton is expected to play the same role as the B 6 potential does for the NS5-brane [9]. More generally, one expects from T-duality the appearance of further mixed-Young tableaux fields as exotic duals [13,11] of the usual gauge potentials [15], as further clarified in [16], together with associated 'exotic' branes [17]. We find that all expected fields are indeed described by the dual DFT.
The results of previous studies suggest that all dual fields can be organized into a 4-index antisymmetric tensor under O(D, D). A first attempt to introduce a 4-index antisymmetric tensor into the DFT action (together with sources) was performed in [18] in the formulation of [18,19]. Moreover, in [20] it was argued that the duality relation between the embedding tensor θ M N P of lower dimensional supergravity and a (D − 1)-potential D (D−1),M N P can be uplifted to higher dimensions by introducing mixed-symmetry dual potentials (in particular, relating the so-called Q-and R-fluxes to (8,2) and (9, 3) mixedsymmetry tensor fields, respectively) and that these mixed-symmetry potentials can be encoded in an antisymmetric 4-index tensor of O(D, D).
Starting from the linearization of the DFT action, written in terms of the linearized frame field that reads h AB = −h BA , we apply the standard procedure of obtaining the dual theory, introducing Lagrange multiplier fields that impose the Bianchi identities for the generalized anholonomy coefficients (sometimes referred to as generalized fluxes). This naturally leads to a 4-index antisymmetric field D ABCD , but also to a field D AB in the antisymmetric 2-tensor representation and a field D in the singlet representation, in the following called D-fields.
where F ab = 2∂ [a A b] , one moves to a first-order formulation where F ab is an independent field, and the Bianchi identity is imposed by introducing a Lagrange multiplierÃ a , This action is gauge invariant under δÃ a = ∂ a Λ, δF ab = 0. Varying w.r.t.Ã a one obtains the Bianchi identity ∂ [a F bc] = 0, which can be solved in terms of the Maxwell potential, giving back the original Maxwell theory. Conversely, one can solve for F in terms ofÃ a to obtain the duality relation whereF ab = 2∂ [aÃb] is the dual field strength. Insertion into the action leads to the dual Maxwell action forF ab . By applying the same procedure in any dimension and for any p-form potential A p , one obtains a dual (D − p − 2)-form potentialÃ D−p−2 , whose gauge paramenter is a (D − p − 3)form, δÃ D−p−2 = dΛ D−p−3 .
(2. 4) In order to set the stage for the comparison with the dualization in DFT, we will often consider the Hodge duals of the potentialÃ D−p−2 and the gauge parameter Λ D−p−3 . The corresponding field is denoted byÃ p+2 and the parameter by Λ p+3 . The field strengths and the gauge variation then take the divergence form F a 1 ...a p+1 = ∂ bÃ ba 1 ...a p+1 , δÃ a 1 ...a p+2 = ∂ a Λ aa 1 ...a p+2 , (2.5) while the corresponding first-order action reads For instance, consider a 2-form b 2 in D dimensions with field strength H abc = 3∂ [a b bc] . Starting from the standard action we pass to a first order action with a fully antisymmetric 4-tensor D abcd and 3-form H abc as independent fields, The equation for D abcd gives the Bianchi identity for H, ∂ [a H bcd] = 0, while the equation for H gives the duality relation The action and field equations are invariant under the gauge transformation where Σ eabcd is completely antisymmetric. The more familiar form of the duality relation is obtained by passing to the Hodge-dual (D − 4)-form 11) in terms of which (2.9) reduces to the standard duality relation between the (D − 3)-form field strength of this (D − 4)-form potential and H. Alternatively, defining the above duality relation reads H abc =H abc . The 'field strength' G abc in the above equation will appear naturally in Section 6. The equations of motion and Bianchi identity for the dual field are then swapped with respect to the original variables:

The dual graviton
We now repeat the same analysis for the dual of the D-dimensional graviton at the linearised level, following [2,26]. We write the linearized Einstein-Hilbert action for the vielbein fluctuation h a|b (including the antisymmetric part, as indicated by the bar) as We now pass to a first order action by adding the Lagrange multiplier D abc d ≡ D [abc] d to impose the Bianchi identity, Varying w.r.t. D abc d and f ab c , respectively, gives The first equation implies locally that f takes the form (2.16). The second equation is then the duality relation between the graviton, contained in h a| b , and the dual graviton, contained in D abc d . From this duality relation we may recover the original (linearized) Einstein equations (2.18) by acting with ∂ a and using that the right-hand side gives zero by the 'Bianchi identity' ∂ d ∂ a D dab c ≡ 0.
Conversely, we can express the theory in terms of the dual variables. We first note that in terms of the 'field-strength' for the dual graviton, the duality relation is equivalent to where we reinserted the explicit potentials in the last step. Inserting now this expression for f in terms of D into (2.19) one obtains the dual action for D. Let us discuss the physical content of the dual theory in a little more detail. To this end we decompose where D (tr)abc d is traceless and D ′ab = 1 (D−2) D abc c is the trace part. In order to further elucidate the representation content, consider the 'Hodge-dual' field whose irreducible GL(D) representations are given by It is easy to see that the traceless potential D (tr)abc d in eq. (2.23) corresponds to the (D − 3, 1) mixed Young-tableaux representation, while D ′ab corresponds to the totally antisymmetric (D − 2). It turns out that the totally antisymmetric representation is pure gauge. Indeed, the gauge invariance of the linearized Einstein-Hilbert action (2.15) under diffeomorphisms and local Lorentz transformations elevates to a gauge invariance of the master action (2.19), acting on the fields (2.23) as Due to this Stückelberg invariance, the field D ′ab drops out of the action upon insertion of (2.22) into (2.19), leaving a two-derivative action for the physical dual graviton in the (D − 3, 1) Young tableau representation. 4 The D-fields also possess gauge transformations that leave the 'field strength' G a bc and hence the action and duality relations invariant,

Dual graviton and dilaton
We now consider the dual graviton and dilaton together at the linearized level. We first consider a canonically coupled scalar (i.e. in Einstein frame) with Lagrangian L = R − where the antisymmetric D (E)ab is the Lagrange multiplier whose equation of motion yields the Bianchi identity This implies locally f (E) a = ∂ a ϕ, from which we recover upon reinsertion into (2.29) the original scalar theory. Alternatively, varying w.
gives the duality relation and eliminating f (E) a accordingly from (2.29) yields the theory for the dual dilaton D (E)ab (or, equivalently, for the (D − 2)-form potential). The action and duality relations are invariant under the gauge tranformation

32)
4 It should be emphasized that while the standard Einstein-Hilbert action can be written entirely in terms of the symmetric h (ab) and the dual action entirely in terms of the irreducible (D − 3, 1), the dualization requires the presence of an antisymmetric part, either h [ab] or D ′ab , since in the master action (2.19) or the duality relations (2.20), local Lorentz invariance allows us to set only one to zero, not both. 5 The superscript E refers to the Einstein-frame.
where Σ abc is fully antisymmetric. The above dualization of a scalar was completely decoupled from the dualization of gravity. For dualization in string frame, however, this picture changes significantly in that it will be the trace of the field D abc d (c.f. the previous subsection) that becomes the dual dilaton, while the analogue of D (E)ab will be pure gauge, transforming with a shift under local Lorentz transformations.
We start from the action with Lagrangian L = e −2φ (R+4(∂φ) 2 ) for the graviton-dilaton sytem, whose linearization yields where The f ab c and f a satisfy the following Bianchi identities: (2.37) Varying w.r.t the fundamental fields D abc d , D ′ab , f a and f ab c , respectively, one obtains where we introduced g ab c and g a for convenience. The equations of motion and Bianchi identities for the dual system are then B.I's: In order to further analyze the content of these equations it is useful to decompose D as follows where D (tr)abc d is traceless andD ab =D [ab] the trace part. The equations of motion for the components then read Note that the D ′ab 's dropped out, which means that they are subject to a Stückelberg symmetry. From the duality relations (2.45) and the split (2.47) it is easy to obtain the usual duality relation between the dilaton and the dual dilaton: We observe that D ′ab disappears and the fieldD ab (the trace of D abc d ) is the dual dilaton, which is the opposite of the situation in Einstein frame.
We close this subsection by discussing the gauge transformations for the D-fields. The duality relations (2.40) and the master action are invariant under local Lorentz transformations with D ′ab and D abc d transforming as W.r.t to the decomposition (2.47) this implies in particular δ Λ D (tr)abc d = 0 and δ ΛDab = 0, implying that the physical dual graviton and the dual dilaton are invariant. The D-fields also possess gauge transformations that leave the 'field strengths' G a bc and g a (and hence the duality relations and action) invariant, (2.51) The gauge parameters satisfy Σ eabc d = Σ [eabc] d and Σ abc = Σ [abc] . One may decompose into traceless and trace parts in order to read off the transformations for D (tr)abc d andD ab .

Exotic Dualization of Kalb-Ramond field
In this section we will discuss the dualization of a 2-form gauge potential ('the B-field') into exotic mixed Young tableau fields. We first review general aspects of such mixed Young tableau gauge fields and then turn to a master action that can be used to dualize the B-field into such a tensor, provided extra fields are included. These fields are quite unusual in that they are not auxiliary (they cannot be eliminated algebraically) nor pure gauge, yet they do not add to the propagating degrees of freedom.

Generalities of (D − 2, 2) Young tableaux gauge fields
We start by discussing general aspects of gauge fields in mixed Young diagram representations; see [10,11] for a systematic treatment and [12] for the construction of invariant actions. Here we specialize to the (D − 2, 2) Young diagram representation, for which the gauge field is subject to There are two types of gauge parameters, µ ∈ (D − 3, 2) and λ ∈ (D − 2, 1), acting as 6 These gauge transformations preserve the algebraic constraints on B. We can define a gauge invariant curvature, starting from the first-order generalized Christoffel symbol which is invariant under µ transformations and satisfies the Bianchi identities As common for Young tableau fields with more than one column, this first-order object is not fully gauge invariant (it is analogous to the Christoffel symbols), because under λ transformations we have A fully gauge invariant curvature is the Riemann-like tensor obtained by taking another derivative and antisymmetrizing over three indices, This Riemann tensor satisfies the Bianchi identities and hence lives in the (D − 1, 3) Young diagram representation. Naively, one would now impose the Einstein-type field equations that set to zero the generalized Ricci tensor R a 1 ...a D−2 d ,bcd , but it turns out that a theory with these field equations is actually topological. To see this note that these field equations imply vanishing of the double-trace of the Riemann tensor, which by the equivalence implies vanishing of the full Riemann tensor and hence that the field is pure gauge. However, we can impose weaker field equations that do lead to propagating degrees of freedom, setting to zero the triple-trace of the Riemann tensor, Note that these are the same number of equations as for the conventional dual of a 2-form (D − p − 2 = D − 4), but now these are equations for the (D − 2, 2) gauge field. (Such dualities have been discussed by Hull in [13]; see also [14] for similar exotic dualizations.) This also proves that there can be no action principle implying (3.9) for the (D − 2, 2) gauge field alone -simply because variation w.r.t. the (D − 2, 2) field would yield more equations. However, one can write an action that implies this field equation at the cost of introducing more fields (that are not pure gauge), which also serves as a master action proving the equivalence with the standard 2-form action, as we will now discuss.

Master action
In order to construct this master action we follow [21] and write the standard action for the Kalb-Ramond field up to total derivatives as We can then replace it by the first-order action where the fields have the symmetries Note that, as usual for master actions, these fields do not live in irreducible representations. The above action may seem like a rather unnatural rewriting of a 2-form theory, but we will see in sec. 6 that, in the appropriate sector, DFT reproduces precisely such an action. This action is invariant under the gauge transformations Let us now verify the equivalence with the second-order action. We vary w.r.t. D and Q, respectively, to obtain (3.14) Reinserting the solution of the first equation into the action we recover the original secondorder action (3.10). Equivalently, at the level of the equations of motion, we can act on the second equation with ∂ a to obtain the Bianchi identity which becomes the standard second-order equation for b ab . Thus, the first-order action is on-shell equivalent to the second-order action. In order to determine the dual theory, we have to use the second equation in (3.14) (the duality relation) and solve for Q in terms of D, which upon reinsertion into (3.11) yields the dual action for D, Variation w.r.t. D yields the second-order equation which is equivalent to the result obtained from (3.16) by taking a curl and using the Bianchi identity ∂ [a Q b]|cd = 0.
In the remainder of this section, we will analyze the dual theory in a little more detail. We first decompose D into its irreducible representations: where we decomposed at the right-hand side into traceless tableaux (indicated by a tilde) and the trace parts. Thus, the decomposition (into not yet irreducible representations) reads whereD is fully traceless, corresponding to the first three representations in (3.19), and C a|b is a general 2-tensor (with antisymmetric and symmetric parts), corresponding to the last two representations.
We will now show that the duality relations imply the correct equations for the (D−2, 2) field. In order to simplify the index manipulations we specialize to D = 4, which shows already all essential features, and for which the conventional dual to the B-field is a scalar and the exotic dual is a (2, 2) tensor. In this case we can decompose D as where B is the 'Hodge-dual' form of the tracelessD in (3.20) and hence lives in the (2, 2) Young tableau. Moreover, we have redefined the general 2-tensor for later convenience, The Σ gauge symmetries can be decomposed as follows so that we can writeΣ where λ ∈ (2, 1) and ξ is a new vector gauge parameter. Applying the gauge transformations (3.13) to (3.21) and using this decomposition of the gauge parameter one finds the following gauge transformations for the component fields: The transformation in the first line is precisely the expected gauge transformation of a (2, 2) gauge field, c.f. (3.2), while the symmetry parametrized by µ in (3.2) trivializes in D = 4 because there is no (1, 2) Young tableau. Note that the extra field C a|b transforms under the gauge symmetry parametrized by λ ab,c . The duality relation (3.16) in terms of B and C reads with the generalized Christoffel symbol (3.3). It is an instructive exercise to verify the gauge invariance of this equation: Under 'b-field gauge transformations' with parameter ξ a the left-and right-hand sides are not invariant, but their respective variations precisely cancel. The right-hand side is manifestly invariant under the ξ a transformations, while under λ transformations the variations of the two terms on the right-hand side cancel. We next show that the duality relation implies as integrability condition the desired field equation for the (2, 2) field. To this end we act on (3.26) with ǫ abij ∂ a , for which the left-hand side gives zero, and one obtains Now summing over i, c and j, d, the second term depending on C drops out, leaving Thus, we obtained the expected field equation (3.9) for D = 4, which proves that the (2, 2) gauge field propagates the single degree of freedom of the b-field in D = 4.

Dual action
Let us finally determine and analyze the Lagrangian in terms of the dual fields, obtained by substituting (3.21) into (3.17), (3.29) It is amusing to write this in a slightly more geometric form by defining the generalized 'Einstein tensor' which satisfies the Bianchi identities and in terms of which the action reads Note that decomposing C into symmetric and antisymmetric parts, C a|b = s ab + a ab , with s ab ≡ s (ab) , a ab ≡ a [ab] , the generalized Einstein tensor becomes in terms of the standard 3-form curvature h abc ≡ 3∂ [a a bc] and the (linearized) Einstein The above Lagrangian can then be written as Curiously, one obtains the conventional (linearized) Einstein-Hilbert term for s ab plus the standard kinetic term for a ab , both multiplied by an overall factor of −2. These wrongsing kinetic terms for a 'graviton' and a 'Kalb-Ramond field' naively would lead one to conclude that this theory propagates a ghost-like spin-2 mode and (in D = 4) a scalar mode. However, since the action is not diagonal and since these fields are subject to larger gauge symmetries parameterized by λ ab,c , there is no conflict with the equivalence to a single scalar mode, which is guaranteed by the construction from a master action. As a consistency check, let us verify that this action indeed implies the expected field equation for the (2, 2) field. Varying (3.32) w.r.t. B ab,cd and C a|b , respectively, yields ∂ e Γ e ab,cd − R ⋆ ab,cd (C) = 0 , where denotes the projection onto the (2, 2) Young diagram representation, 7 and we defined the analogue of the linearized Riemann tensor for C a|b and its dualization This Riemann tensor satisfies the Bianchi identity R [abcd] = 0, which in turn implies that the double trace of R ⋆ ab,cd vanishes (note, however, that R [abc]d generally is non-zero because C carries an antisymmetric part). As a consequence, taking the double trace of the first equation in (3.36), the R ⋆ term drops out, implying the required field equation R abc ,abc = 0, precisely as in (3.28). The (2, 2) projection of the dual Riemann tensor in (3.36) plays a role analogous to the Weyl tensor in Einstein gravity (where it is left undetermined by the field equations and hence encodes the propagating graviton degrees of freedom). Here, on the contrary, the tensor R ⋆ ab,cd is fully determined by the (2, 2) gauge potential, in agreement with the non-propagating nature of C a|b . 7 Explicitly, acting on a tensor X ab|cd that is antisymmetric in each index pair, this projector reads

Dualizations in Linearized DFT
In this section we discuss the relations between dual and standard fields in Double Field Theory (DFT), using linearized DFT in the frame formulation [5,7,18]. We will add Lagrange multipliers (denoted as D -fields in the following) to the linearized DFT action in order to enforce the Bianchi identities. This will allow us to obtain duality relations between the conventional fields and the D-fields and, as integrability conditions, second order differential equations.

Linearized DFT in frame formulation
The where η ab and ηāb are two copies of the flat D-dimensional Lorentz metric diag(− + · · · +), and the relative sign between them is so that the overall signature is compatible with the (D, D) signature of η M N . A different but equivalent form of the constraint is given by choosing the flattened metric so that it takes the same form as the O(D, D) metric, where we denoted the frame field by E A M to indicate that it satisfies a different constraint. Due to this constraint, E A M is a proper O(D, D) group element. The flat indices split as A = ( a , a ) and, therefore, in this formalism one has to carefully distinguish between upper and lower indices. The tangent space indices are raised and lowered with η AB or G AB , depending on the formalism.
The generalized metric encoding metric g and b-field can be defined conventionally in terms of the frame field. For instance, in the formalism based on (4.4), we define the where η ab and η ab are again two copies of the flat Lorentz metric, in terms of which the generalized metric can be written as In the following we use the perturbation theory for both formalisms, with frame fields subject to either (4.3) or (4.4), because each is more convenient for different purposes. In the remainder of this section we discuss the formalism based on (4.4), using the conventions of [18], while the formalism based on (4.3) will be discussed and applied in sec. 5.
We now discuss the frame-like perturbation theory, whose details have been developed in [22] for flat and curved backgrounds. Here we consider perturbations around a constant background, writing The constraint (4.4), which requires E A M to be O(D, D) valued, implies to first order in the fluctuation h AB + h BA = 0. We thus assume h AB to be antisymmetric. 8 Moreover, in the following we denote the linearization of the dilaton by d and its background value byd. The linearized theory is naturally written in terms of generalized coefficients of anholonomy, also known as generalized fluxes [18,19], which are defined as with the flattened (doubled) derivative Note that in DFT we impose the 'strong constraint' ∂ M X ∂ M Y = ∂ M ∂ M X = 0 for any fields X, Y , which then implies D A D A = 0 acting on arbitrary objects (which we will sometimes abbreviate as D 2 = 0). It is then easy to verify that the above coefficients of anholonomy satisfy the Bianchi identities (4.10) Conversely, it is straightforward to prove, using the Poincaré lemma and the strong constraint D A D A = 0, that the general solution of these equations is given by (4.8).
Let us now turn to the linearized DFT action, which takes the form whereF ABC is defined as: with the short-hand notation The tensorsS and S satisfy the following identities: (4.14) The action (4.11) is invariant under infinitesimal generalized diffeomorphisms (with the generalized coefficients of anholonomy being invariant to first order) and local double Lorentz transformations δ Λ h AB = Λ AB , with infinitesimal parameter Λ AB satisfying The equations of motion following from the linearized DFT action (4.11) for h AB and d, respectively, are given by

Master action and duality relations
We now pass to a first-order or master action as in previous sections, promoting F A and F ABC to independent fields and introducing (totally antisymmetric) Lagrange multipliers D ABCD , D AB and D that enforce the Bianchi identities. The action thus reads Varying w.r.t. the fundamental fields D ABCD , D AB , D, F ABC and F A , respectively, we obtain the field equations D [A F BCD] = 0 , (4.20) With the first three equations we recover the Bianchi identities, which can be solved as in where Σ ABCDE = Σ [ABCDE] and Σ ABC = Σ [ABC] . The D-fields also transform under double Lorentz transformations. Using (4.16) in the above duality relations, one finds

Dual DFT
Let us now investigate the equations of motion for the theory in terms of the dual D-fields. These are obtained from the Bianchi identities (4.10) and the duality relations (4.23)-(4.24). First, we need to solve the duality relations for the coefficients of anholonomy in terms of the dual D-fields, which yields, using eq. (4.14), In order to illuminate further these equations for the dual D-fields, let us introduce the following field strengths: and which are invariant under the Σ-transformations (4.26). In terms of these field strengths the duality relations take the following simpler form: Finally, defining G ABC ≡S ABC DEF G DEF and G A ≡ S A B G B , the second-order equations (4.29)-(4.31) for the dual fields take exactly the same form as the Bianchi identities for the original fields. Our final form of duality relations between fluxes and dual fluxes is then (4.35) The set of equations for the original and dual system is summarized in Table 1.

Geometric form of dual DFT action
In this section we elaborate on the geometric form of the dual DFT action. We first present a master action in terms of connections that, in a sense, is complementary to that presented in sec. 4, but which leads to equivalent results. Finally, we determine the dual action and write it in a geometric form that is completely analogous to the dual action for the exotic duals discussed in sec. 3.

DFT action in connection form
In order to define the master action in a (semi-)geometric form, let us first review the linearized frame-like geometry of DFT, based on a frame field E A M , where the flat indices split as A = (a,ā). Since the frame field is subject to (4.3), expansion about a constant background, leads to the following first-order constraints on the fluctuations where Λ AB = diag(Λ ab , Λāb), and therefore while the dilaton transforms as δd = − 1 2 (∂ a ξ a + ∂āξā) .  δω abc = ∂ a Λbc , δωā bc = ∂āΛ bc , In particular, the connections are fully invariant under generalized diffeomorphisms. The above connections satisfy the Bianchi identities ∂ a ω a + ∂āωā = 0 , This is a rather extensive list of identities, but except for the first one they are all consequences of the algebraic Bianchi identity for the full Riemann tensor, R [ABC]D = 0, see [28], and are also equivalent to (4.10). We now give invariant curvatures in order to define the dynamics of linearized DFT. There is a linear generalized Riemann tensor, which, however, does not have a non-linear completion. The linearized (generalized) Ricci tensor (which is not the trace of the above Riemann tensor) reads R ab ≡ −∂ c ωb ac + ∂b ω a ≡ −∂cω abc + ∂ a ωb , (5.10) where the equivalence of the two definitions follows from the fourth Bianchi identity in (5.8). The explicit expression in components reads where ≡ ∂ a ∂ a ≡ −∂ā∂ā. As it should be, the pure gauge degrees of freedom dropped out. Also note that there are differential Bianchi identities relating (5.9) to (5.10), The linearized scalar curvature is where we have given the explicit component expression in the last step. Finally, the linearized DFT action in terms of the connections reads (5.14) whose general variation reads δL = 4δh ab R ab − 8δd R. Let us note that, upon inserting (5.6), the two lines in the above action actually give the same result, by virtue of the strong constraint and the relative sign between them, but for our present purposes this action is convenient because it treats barred and unbarred indices on the same footing.

Master action
We now give a first-order master action that can be used to define the dual theory and in which the connections are promoted to independent fields, in analogy to previous sections. Apart from that, the approach is complementary to that used in previous sections in that the dual fields do not enter the master action as Lagrange multipliers but rather emerge upon 'solving' the field equations by reinterpreting them as Bianchi identities. This approach is of course fully equivalent to that used before (the difference being whether the fields or their duals enter the master action that serves as the starting point), but it is reassuring to confirm explicitly that both procedures give the same result.
We now treat the connections as independent fields and replace the linearized DFT action (5.14) by the first-order action The field equations for the ω determine them in terms of the physical fields as given in (5.6), so that reinserting into the action we recover (5.14). On the other hand, varying with respect to d, h ab , h ab and hāb, respectively, we obtain ∂āωā − ∂ a ω a = 0 , −∂c ω abc − ∂ c ωb ac + ∂ a ωb + ∂bω a = 0 , Expressing ω in terms of the physical fields, the first two equations give the DFT equations R ab = 0 and R = 0, while the last two equations are the second and third Bianchi identity in (5.8).
In order to determine the dual theory we interpret now all four of the equations (5.16) as Bianchi identities and solve them in terms of dual fields. We proceed hierarchically, starting with the first equation, which can be solved as (5.17) with Dāb and D ab antisymmetric, D bā unconstrained and a singlet D. This result can be obtained as follows. First, the non-singlet terms follow from the standard Poincaré lemma, writing the equation as ∂ A Ω A = 0 for Ω A ≡ (−ω a , ωā), which implies Ω A = ∂ B D AB for antisymmetric D AB , whose components give the above D fields. The only subtlety is that the derivatives are subject to the strong constraint, which allows for the singlet term that drops out by ∂ā∂ā = −∂ a ∂ a . Thus, (5.17) is the general solution of the first equation in (5.16).
Next, we turn to the second equation in (5.16), where we can eliminate ω a and ωā according to (5.17). We first solve the equation for the special case that all these D fields are zero: − ∂c ω abc − ∂ c ωb ac = 0 . where the D fields are antisymmetric in each group of similar indices. Including now the trace connections we need to solve the inhomogeneous equation ∂c ω abc + ∂ c ωb ac = ∂ a ∂cDbc + ∂ a ∂ c D cb + ∂b∂ c D ac + ∂b∂cD ac , (5.20) where we note that the singlet D dropped out. This equation is solved by which can be verified by employing the strong constraint again. Thus, the general solution is given by the sum of (5.19) and (5.21), Finally, we solve the last two equations in (5.16). Inserting (5.17) and (5.22) determines ω [abc] up to solutions of ∂ a ω [abc] = 0, which by the Poincaré lemma are given by ∂ d D abcd for a new totally antisymmetric tensor D abcd . Applying the same reasoning to ω [ābc] introduces the new field Dābcd, and we finally find for the connections in terms of the dual fields, Comparing with the list of Bianchi identities (5.8) we infer that the D fields and Bianchi identities are in one-to-one correspondence. Thus, these fields could be used as Lagrange multipliers to impose the Bianchi identities, confirming the equivalence with the master action procedure discussed in sec. 4. We now turn to the dual gauge symmetries that leave (5.23) invariant and thus describe the redundancies between the D fields. For the two-index fields one finds Note that the dual diffeomorphism parameters Σ a and Σā act on these fields in exactly the same way as the original diffeomorphism parameters ξ a and ξā act on h ab , hāb, h ab and d. For the four-index field we find Thus, exactly as for h ab and hāb, these fields are pure gauge.

Geometric action for dual DFT fields
Let us now insert (5.23) into the master action (5.15) in order to obtain the action for the dual D fields. The terms involving the original fields drop out because these fields enter linearly, multiplying constraints that have been solved in terms of the D fields. The second-order action therefore reads with the connections given by (5.23). This takes precisely the same form as (5.14), except that the overall sign has changed. The computation of inserting (5.23) is simplified by using that the dependence of ω on D ab , D ab , Dāb and D is precisely analogous to the expressions in terms of the original fields, up to the following identifications, and an overall sign for the connections with unbarred Lie algebra indices, which is irrelevant since the connections enter the action quadratically. A direct computation yields the explicit form of the dual Lagrangian, L = − 1 2 ∂dDbcd ,a ∂ēDbcē ,a − ∂dDbcd ,a ∂ e D ea,bc − 1 2 ∂ d D da,bc ∂ e D ea,bc + 1 2 ∂ d D bcd,ā ∂ e D bce,ā + ∂ d D bcd,ā ∂ēD bc,āē + 1 2 ∂dD bc,ād ∂ēD bc,āē − 3 2 ∂dDābcd ∂ēDābcē − ∂dDābcd ∂ e Dābc ,e − 1 6 ∂ d Dābc ,d ∂ e Dābc ,e DFT (D ab , D) .

(5.30)
Note that in the last line we encounter the standard linearized DFT Lagrangian L (2) , but for D ab and D, with the 'wrong' overall sign, in complete analogy to the mixed Young tableau action discussed in sec. 3. Also in perfect analogy to that discussion is that this wrong-sign kinetic term does not indicate the presence of ghosts, for the action is not diagonal. Rather, the off-diagonal term is proportional to the linearized Riemann tensor (5.9), but expressed in terms of D ab . Thus, the Σ a and Σā transformations are manifest symmetries of this action, while the invariance under the remaining dual diffeomorphisms (5.26) can be verified by a direct computation. Also note that the fields Dāb and D ab dropped out, as it should be in view of the Stückelberg-type Lorentz invariance (5.27).
We close this section by discussing two of the D-field equations, because they exhibit an intriguing structure. Varying w.r.t. D ab and D we obtain For the first equation neither the left-hand side nor the right-hand side are dual diffeomorphism invariant under transformations with parameter Σ ab,c and Σāb ,c , but their variations precisely cancel against each other. The field equation for D ab,cd reads where we defined Thus, intriguingly, the equation takes the form of a second-order duality relation, relating the (linearized) Riemann tensor to a 'dual' Riemann tensor. As above, both sides are not separately invariant under dual diffeomorphisms with parameter Σ ab,c and Σāb ,c , but the full equation of course is, as it should be and as may be verified by a quick computation.

Comparison of results
In Section 4 we have shown that at the linearized level the DFT equations and Bianchi identities for the fluxes F ABC and F A arise from first order duality equations given, for instance, in eq. (4.35), relating these fluxes to the dual fluxes G ABC and G A . The dual fluxes are defined in terms of the field strengths G ABC of the dual potentials (the D-fields) in eqs. (4.32) and (4.33). The field equations and Bianchi identities for the fields and the dual fields are listed in Table 1. The aim of this section is to show that if one restricts all DFT fields to only depend on x, i.e. if one sets∂ µ Φ = 0 for any DFT field Φ, one recovers the previous results of dualization: the standard dualities between the 2-form and the (D − 4)-form and between the graviton (plus dilaton) and the mixed-symmetry (D − 3, 1) potential discussed in sec. 2, and the exotic duality between the 2-form and the mixed-symmetry (D − 2, 2) potential discussed in sec. 3. The dual potentials introduced in sec. 4 are D ABCD , D AB and D. Upon breaking O(D − 1, 1) × O(D − 1, 1) to the diagonal subgroup, the field D ABCD can be decomposed as while the field D AB decomposes as When reducing to x-space we use, by a slight abuse of notation, the same symbols for the components of the DFT D-fields and the supergravity D-fields. The identification uses the ordering of the indices as given above to match the results of the previous sections.
The same applies for the components of G ABC . We make an exception, in the following subsection, for the identification of the components of D AB and D with the ones in x-space: the convention being that x-space fields carrying a prime can be gauged or redefined away. If one inserts the above identifications into eq. (4.19), one recovers the first order actions of Section 2. In particular, the fields D abcd , D abc d and D ′ab are precisely the potentials that we introduced in Section 2 when we performed the standard dualization for the 2-form and the graviton plus dilaton system. This requires that, in x-space, the fields D ′a b and D ′′ can be redefined away and/or are irrelevant for the analysis. We will also see that D a bcd , D abcd and D ab trivialize in x-space frame.
One can also recover the duality relations for each field by performing the decomposition directly in the duality relation (4.35). We first identify the components of F ABC in x-space as: which at this stage are just labels for the components of the F flux. As we will see, H abc , f ab c and f a play the same role as in in Section 2, and we will discuss later Q a bc , Q a and R abc , which are related to non-geometric fluxes. Note that because of the presence of the tensorS ABCDEF in the definition of G ABC in terms of G ABC , eq. (4.35) relates a given component of F ABC to different components of G ABC and thereby to different components of the dual potentials. This has to be understood as follows: if one turns on a particular component of the flux F ABC , eq. (4.35) still gives equations for all the dual potentials. The equations for the dual potentials dual to the vanishing fluxes will furnish algebraic relations among the different components of G ABC , and after reinserting these relations into the duality relation for the non-vanishing fluxes one finds that this is dual to a specific component of G ABC suitably antisymmetrized. This will also be discussed in each case in the remainder of this section, which is organized as follows. In the first subsection we will show how from DFT one recovers the standard dualizations of Section 2, while in the second subsection we will show how the exotic dualization of Section 3 is also contained in DFT. Finally, in the third subsection we will briefly discuss the remaining dual fields, which are related to non-geometric fluxes such as the R-flux.

Standard duality relations for the 2-form and graviton plus dilaton
The truncation of the action given in eq. (4.19) to x-space with only either the H-flux or the f -flux turned on straightforwardly reproduces the field theory analysis of Section 2. In the case of the H-flux, only the component D abcd of D ABCD appears in the action, and one immediately recovers eq. (2.8). In the case of the f -flux, one turns on only the component D abc d in D ABCD and D ′ab in D AB to recover precisely the action in eq. (2.37). The analysis performed in Section 2 showed that D ′ab is pure gauge while D abc d describes both the dual of the graviton and the dual of the dilaton.
As anticipated at the beginning of this section, a more careful analysis is required if one wants to perform the truncation at the level of the duality relations. In the case of the H-flux, the duality relation (4.35) simply gives H abc = G abc , with the other components of G ABC vanishing. In terms of G ABC this gives In this equation both G abc and G ab c occur, but one has to take into account also the equation for the vanishing dual flux G a bc , which gives implying the algebraic relation Upon inserting this relation into eq. (6.5) one obtains which is in agreement with (2.12), identifying G abc =H abc and using the definition of G abc given in eq. (4.32). We now perform the same analysis for the graviton-dilaton system. Turning on only the fluxes f ab c and f a in eq. (4.35) we must recover eq. (2.45), where G ab c is identified with g ab c and G a with g a . In terms of G ABC , one has The two components G abc and G a bc that occur in this equation are related by the condition that the dual flux G abc vanishes, which yields the relation Inserting this into eq. (6.9) one obtains which precisely reproduces eq. (2.45) by using (4.32). It is also straightforward to show that G a coincides with g a defined in (2.45) after using eq. (4.33).

Q-flux dualization from DFT
We now consider the truncation to x-space of the DFT dualization for the Q-flux component in (6.4) and show that it reproduces the exotic dualization of the 2-form discussed in Section 3. We start from the first order action (4.19), specialized to the Q-flux components, and reduce to x-space, where the fields D ab cd ≡ D [ab] [cd] , D a b , D and Q a , Q a bc are independent, and we dropped the primes relative to (6.3). The field equations for the D-fields read which are the Bianchi identities (4.10), reduced to x-space and specialized to the components Q a bc and Q a . The solution of these equations is and we will see in the following that the constant term is irrelevant. The field equations for Q a and Q a bc yield the duality relations 15) which are equivalent to the duality relations following from (4.23) and (4.24) upon specializing to the Q-fluxes.
Comparing with the master action discussed in sec. 3, we observe that here we have Lagrange multiplier fields, D a b and D, which have no analogues in that previous analysis, but we will now show that these fields are irrelevant. We first note that (6.12) is invariant under the gauge transformations with local parameter χ with δ χ Q = 0. These act as a Stückelberg symmetry on D. Thus, we can gauge this field to zero. 9 Equivalently, we can express the action directly in terms of the gauge invariant objects which yields As expected, the singlet D field dropped out. The field D a b cannot be eliminated similarly by a gauge symmetry. Rather, its own field equation yields the second of the Bianchi identites in (6.13), and back-substituting their solution (6.14) into the action (6.18) gives the free Kalb-Ramond action for the b-field, which at the linearized level is equivalent to the 'β-supergravity' for the bi-vector field β ab ≡ b ab (with the indices raised by the flat Minkowski metric) [27]. Note, in particular, that the constant term in (6.14) contributes to the Lagrangian only an irrelevant constant and a total derivative term. Therefore, it is physically equivalent to set the constant to zero, in which case Q a = Q b ba and the second and third Bianchi identity are no longer independent but are traces of the first one. Thus, on-shell the above action is equivalent to the same action with Q a = Q b ba and with the only Lagrange multiplier being D ab cd , enforcing the first Bianchi identity in (6.13). This action is then manifestly equivalent to the master action (3.11) discussed in sec. 3. 10 Thus, we have shown that in the Q-flux sector the DFT dualization reduces to the exotic dualization of the B-field into a mixed-symmetry potential with a (D − 2, 2) Young tableau.

The R-flux
We now consider the R-flux contribution of (6.4) in the truncation of the master action (4.19) to x-space. The action reduces to where D a bcd = D a [bcd] and D ′ ab = D ′ [ab] . Note that the field D ′ ab can be absorbed into the trace of D a bcd . The equations for the dual potentials in this case simply imply that R abc has to be constant and hence that in this sector the fields carry no degrees of freedom. This is consistent with the form of the R-flux in x-space at the non-linear level: whose linearization vanishes for vanishing β background. The duality then implies that the dual flux G abc also vanishes. Finally, let us also note that the field D abcd disappears from the action in x-space since it couples to a Bianchi identity for the R-flux that explicitly contains a derivative∂ µ with respect to the dual coordinate. The field D abcd can be written as a (10, 4) gauge field in D = 10 by using the epsilon tensor, as can be deduced by writing its gauge transformation from eq. (4.26) and keeping only x derivatives. On the other hand, in an O(D, D) frame in which we take all the fields to depend only on the coordinatesx, D abcd would become the 'standard' dual of the field β since the R-flux takes the form which plays precisely the same role as the H-flux in x-space. An analogous inversion of roles also holds for all other fields, as is guaranteed by the O(D, D) invariance of the action (4.19). We summarize this in Table 2:

Conclusions and Outlook
In this paper we have determined the dualization of double field theory (dual DFT) at the linearized level, which captures in addition to the conventional dual fields in D = 10 string theory (the 6-form dual to the Kalb-Ramond 2-form and the 8-form dual to the dilaton) fields in mixed-Young tableaux representations, such as the dual of the graviton and an exotic dual of the 2-form, plus additional fields. The dual fields can be organized into a totally antisymmetric 4-tensor under the T-duality group O(D, D), as suggested by previous studies, but it turns out that defining an O(D, D) covariant master action (and, consequently, an action for the dual fields) requires extra fields. A careful analysis shows, however, that reducing the dual DFT to the physical spacetime yields precisely the expected dual theories. In particular, we analyzed the exotic dualization of the 2-form, following the strategy introduced in [21], which is illuminating because it shows that, besides the dual (D − 2, 2) gauge potential, extra fields are needed that carry the representations and gauge symmetries of metric and 2-form fluctuations. Consequently, they enter the action with (linearized) Einstein-Hilbert and Kalb-Ramond terms, but due to non-trivial couplings to the dual fields this does not upset the counting of degrees of freedom. Similarly, the dual DFT carries, besides the 4th rank O(D, D) tensor, fields with the same representations and gauge symmetries as in the original DFT and, therefore, they enter the action with the usual (linearized) generalized curvature scalar of DFT. Again, because of the coupling to the dual fields, this does not indicate the presence of unphysical modes, as is also guaranteed by the master action.
This unusual feature may provide important pointers for the full non-linear theory yet to be constructed. In general, there are strong no-go theorems implying, under rather mild assumptions, that there is no non-linear action for a mixed-Young tableau field that is invariant under a deformation of the linear gauge symmetries [3,4]. However, in the O(D, D) covariant framework analyzed here, this problem presents itself in a quite different fashion. Because of the coupling to extra fields (carrying the representations of the original DFT fields), the no-go theorem is not applicable, and hence it may well be that there is a consistent non-linear deformation of the dual DFT action (5.30). For instance, this would require finding a non-linear extension of the field equation (5.32). The linearized DFT Riemann tensor appearing on the left-hand side of that equation by itself does not have a non-linear extension [28] (which in turn is the reason that higher-derivative α ′ -corrections require a deformation of the framework [29]), but it is natural to speculate that a non-linear extension exists which deforms not only the left-hand side but also the right-hand side that encodes the mixed Young tableau fields. 11 Another reason to be optimistic about the existence of a non-linear extension is that in 'exceptional field theory' (the extension of DFT to U-duality groups) dual graviton components are already encoded at the non-linear level [23], which is achieved by means of additional (compensator) fields. The detailed formulation of these theories is somewhat different, however, in that they require a split of coordinates and indices so that the mixed Young tableau nature of the dual graviton is no longer visible. Therefore, the precise relation between the dual formulation presented here and that implicit in [23] remains to be established. Once this has been achieved and/or the full non-linear form of the dual DFT has been constructed, we would have a fully duality covariant formulation of the low-energy dynamics of the type II strings in terms of all fields and their duals, both for the RR sector, for which this was established a while ago [30,31], and the NS sector.
The construction of such a theory would be very important for the description of various types of (exotic) branes. Indeed, exotic branes are non-perturbative string states that are electrically charged with respect to mixed-symmetry potentials. The branes that are charged under the D potentials discussed in this paper have tensions that scale like g −2 s in string frame. While the NS5-brane is charged under the standard potential D abcd , the KK monopole, the Q-brane and the R-brane are charged under the mixed-symmetry potentials D abc d , D ab cd and D a bcd , respectively. The Q-brane solution [17] is locally geometric, while the R-brane does not admit a geometric description. This is clearly in agreement with our findings, namely that one can write down a duality relation in x space at the linearized level for D ab cd but not for D a bcd . Actually, one should also consider non-geometric objects that are charged under the potential D abcd . Upon dimensional reduction, this would give rise to space-filling branes with the same scaling of the tension, which have been classified (see the second ref. in [15]). In general these branes do not have any solution in supergravity, but their existence is crucial for instance in orientifold models.
The 1/2-BPS branes with tension g −2 s satisfy specific 'wrapping rules' [32]: the number of p-branes in D dimensions is given by the number of p+1-branes in D +1 dimensions plus twice the number of p-branes in D + 1 dimensions. This means that these branes 'double' when they do not wrap the internal cycle. As far as the (D − 5)-branes, the (D − 4)-branes and the (D − 3)-branes are concerned, this is expected from the fact that such branes are magnetically dual to the fundamental string, fundamental particles and fundamental instantons, respectively. Therefore, for these branes the wrapping rules are simply the dual of the wrapping rules for fundamental strings, that see a doubled circle, and thus double when they wrap. The fact that all the potentials associated to these branes enter the DFT duality relations discussed in this paper explains why also the (D − 2) and (D − 1)-branes with tension proportional to g −2 s satisfy the same wrapping rules, although they are not dual to propagating fields in x space.
The classification of 1/2-BPS branes in string theory was extended to branes with tension scaling like g −3 s in the string frame in [33]. Such branes are charged with respect to mixed-symmetry potentials that are magnetically dual to the P -fluxes (a prototype of a P -flux is the S-dual of the Q-flux). In [20] it was observed that all such potentials can be collected in the field E M N,α in the tensor-spinor representation of SO (10,10). It would be very interesting to write down a linearized DFT duality relation for such field, precisely as we did for the D fields in this paper. Such field is magnetically dual to the p-form potentials γ a 1 ...ap (with p even in IIB and odd in IIA), that are U-dual to the RR fields (for instance the IIB scalar γ is the S-dual of the RR axion), and group together to form a spinor representation of SO (10,10). The branes with tension g −3 s satisfy different wrapping rules with respect to the g −2 s branes, namely they 'double' both if they wrap and if they do not wrap. The precise DFT duality relation between the potential E M N,α and the potentials γ would give an explanation for this wrapping rule, which is at the moment rather mysterious.