A geometric formulation of exceptional field theory

We formulate the full bosonic SL(5) exceptional field theory in a coordinate-invariant manner. Thereby we interpret the 10-dimensional extended space as a manifold with $\mathrm{SL}(5)\times\mathbb{R}^+$-structure. We show that the algebra of generalised diffeomorphisms closes subject to a set of closure constraints which are reminiscent of the quadratic and linear constraints of maximal seven-dimensional gauged supergravities, as well as the section condition. We construct an action for the full bosonic SL(5) exceptional field theory, even when the $\mathrm{SL}(5)\times\mathbb{R}^+$-structure is not locally flat.

1 Introduction 2 before discussing the geometry of 10-dimensional manifolds with GL(5) + -structure in section 3. We define the relevant GL(5) + -structure, use it to construct the generalised Lie derivative and show that requiring the algebra to close leads to a set of constraints, including the section condition. We also discuss the case when the GL(5) + -structure is locally flat and show how this case reduces to the usual EFT. In section 4 we develop the formalism in order to describe the tensor hierarchy of the EFT, closely following [52,53]. Finally, in section 5 we give the full action, including the "external" seven-dimensional fields, for any GL(5) + -structure, not just locally flat ones. The resulting theory is manifestly coordinate invariant, as well as invariant under generalised diffeomorphisms and external diffeomorphisms. We discuss background-dependence and comment on further work in section 6.

A brief review of SL(5) exceptional field theory
Let us briefly review exceptional field theory, focusing on the SL(5) EFT [12,13,26,29] which, for example, can be used to describe seven-dimensional maximal gauged supergravities. The theory has 10 "extended coordinates" Y [ab] and seven "external coordinates" x µ , with a, b = 1, . . . , 5 and µ = 1, . . . , 7. The bosonic degrees of freedom of the internal sector are described by a generalised metric M ab ∈ SL(5)/SO(5) , (2.1) which can be parameterised by a four-dimensional metric and three-form corresponding to the internal sector of 11-dimensional supergravity, or by a three-dimensional metric, a doublet of two-forms and 3 scalars, the internal sector of IIB [29]. Just as the bosonic degrees of freedom can be unified in the generalised metric M ab , its symmetries, corresponding to diffeomorphisms and p-form gauge transformations, are combined into the generalised Lie derivative Here the parameter of generalised diffeomorphisms, Λ ab , has weight 1 5 under the generalised Lie derivative, so that under a generalised Lie derivative it transforms as In order for these transformations to close into an algebra where the E-bracket is the antisymmetrisation of the generalised Lie derivative, i.e.
[Λ 1 , Λ 2 ] E = 1 2 (L Λ1 Λ 2 − L Λ2 Λ 1 ) , (2.5) one imposes the so-called "section condition" when acting on any pair of fields f and g. There are two inequivalent solutions (i.e. not related by SL(5) transformations) to the section condition [17,29], given by The first, where fields only depend on the four coordinates Y i5 corresponds to 11-dimensional supergravity while the second, with dependence on only three coordinates Y AB , corresponds to type IIB supergravity. One way this manifests itself is that the generalised Lie derivative of the generalised metric generates exactly the diffeomorphisms and p-form gauge transformations of the bosonic fields of 11-dimensional / IIB supergravity. Furthermore, one can define a unique action which is invariant under generalised Lie derivatives. Upon using the appropriate solution of the section condition this reduces to the fourdimensional / three-dimensional internal sector of 11-dimensional / IIB supergravity [12,29].
This can be extended by introducing fields related to the tensor hierarchy of gauged supergravities [54,55], so that the resulting action reduces to the bosonic part of the full 11-dimensional or IIB supergravity [17,19,26]. We will describe this construction in more detail in sections 4, 5 and appendix D, albeit in our geometric formulation.
3 Structure group, generalised Lie derivative and GL(10) connection 3.1 10-manifolds with GL(5) + -structure 3.1.1 GL(5) + -structure In this paper we define a "curved" version of the SL(5) EFT on a 10-dimensional manifold M which admits a GL(5) + -structure. In order to do this, consider first the usual frame bundle π 10 : F 10 M −→ M whose fibre consists of all ordered bases of the tangent bundle and can thus be identified with GL (10). We will label the bases as E ab . The structure group of M can be reduced to GL(5) + if F 10 M/GL(5) + admits a global section and each such global section defines a GL(5) + -structure on M . In other words, a GL(5) + -structure is defined by an equivalence class of frame fields E ab , where u ∈ GL(5) + . Here a, b = 1, . . . , 5 and the pair of indices ab is antisymmetrised, thus denoting the 10-dimensional representation of GL(5) + . In local coordinates we will write the frame fields as with M = 1, . . . , 10 denoting "curved" 10-dimensional indices. A global section of F 10 M/GL(5) + then implies that the transition functions of the frame bundle can be chosen to be GL(5) + -valued.
For each GL(5) + -structure, we can define a principal GL(5) + -bundle π 5 : F 5 M −→ M , whose fibres consist of the equivalence class of frame fields defining the GL(5) + -structure and can thus be identified with GL(5) + . Note that the GL(5) + -structure can also be defined using an invariant tensor. The product 10⊗10⊗10⊗10 of GL(10) contains a singlet in the decomposition under GL(5) + , corresponding to a GL(5) + -invariant tensor, Y ab,cd ef,gh = ǫ abcdi ǫ ef ghi , the "Y -tensor" in the nomenclature of [16]. However, here we will find it more useful to use the equivalence class of frame-fields (3.1) instead of the Y -tensor when discussing the GL(5) + -structure.
Unlike in the usual EFT formulation, we do not assume that our GL(5) + -structure is locally flat. The obstruction to local flatness of this structure introduces a manifest backgroundindependence into our theory. We will return to this point briefly in the discussion 6. Finally, let us emphasise that in general the representative of a GL(5) + -structure cannot be written in the form for some E a i , where i, j = 1, . . . , 5. When this can be done, the GL(5) + -structure is called locally flat.

Fundamental vector fields
Equipped with F 5 M we can construct a 5-dimensional vector bundle, the associated bundle E 5 on which GL(5) + acts in the fundamental representation. The sections of this bundle are "fundamental vector fields" with basis E a , so that we can write Because the vector bundle E 5 has structure group GL(5) + , we can define a "5-dimensional volume-form", η, as a global section of Λ 5 E * 5 . In the E a basis we denote this by where |E| is the determinant of E ab M and ǫ abcde is the alternating symbol which equals the sign of the permutation (abcde). We will often find it useful to use the tensor density ǫ abcde instead of η abcde itself.

GL(10) diffeomorphisms and Killing vectors
We know that GL(10) diffeomorphisms act on tangent vectors via the usual Lie derivative How does a GL(10) diffeomorphism act on a fundamental vector? A general diffeomorphism will not preserve the GL(5) + -structure and thus not act as an automorphism of E 5 . In order to obtain an action on E 5 , we have to restrict ourselves to automorphisms of the GL(5) + -structure, or infinitesimally to GL(5) + -Killing vectors. 3 These satisfy where λ ab cd ∈ gl(5) + . Thus, We can express this in terms of the projector onto the adjoint of GL(5) + , P adj , as The projector is explicitly given by This implies that a Killing vector U M must satisfy (3.12) 3 We would like to thank Daniel Waldram for helpful discussions on this point.
We can now define the Lie derivative of fundamental vector fields with respect to Killing vectors (3.12), in terms of the connection ∇, as follows Here (3.14) denotes the adjoint action on the fundamental representation of GL(5) + . The coefficient in front of P adj and the torsion terms are chosen in order for (3.13) to be independent of the choice of GL(5) + -connection. Let us briefly expand on the connections appearing here. We require the connections in (3.11), (3.12) and (3.13) to be compatible with E ab , i.e. they induce a "spin connection" ω M,a b in order to satisfy the Vielbein principle We have a GL(5) + -connection when so that ω M,a b is a gl(5) + -valued one-form. When ω M,a a = 0 we have a SL(5)-connection, which is what we will make use of in section 3.3.

Generalised Lie derivative and closure constraints
Similar to DF T W ZW [47][48][49], we now define the exceptional field theory on the flattened spaces associated with the GL(10) vielbeine E ab M . 4 Thus, we make use of the anholonomic derivatives and define the generalised Lie derivative acting on V ∈ Γ (E 5 ) as For now we only require τ bc,d a to be constant but otherwise arbitrary, and will determine it soon.
The generalised Lie derivative generates a SL(5) action if τ ab,c c = 0. Otherwise, τ ab,c c generates an additional R + action. In EFT, weighted vectors play an important role. For a vector of weight w, we thus define the generalised Lie derivative This mirrors the form of the genersalised Lie derivative of gauged EFTs [56]. 5 As in EFT, we take the parameters of generalised diffeomorphisms Λ to have weight w = 1 5 under the generalised Lie derivative and ask for the algebra of these generalised Lie derivatives to close, i.e.
This ensures covariance of the expression (3.19). A straightforward but tedious calculation, detailed in appendix A shows that this is achieved when we impose four types of constraints, which we will collectively refer to as the closure constraints. First, we must identify τ ab,c d in (3.19) with the coefficients of anholonomy as follows In addition, we find that τ ab,c d must satisfy the linear and quadratic constraints of seven- where Z (ab),c = Z [ab,c] = 0, S [ab] = 0 and τ (ab) = 0. Note that the embedding tensor here is related to the one in [58],τ ab,c d , bŷ (3.28) Finally, we require a "section condition" for the anholonomic derivatives where the ⊗ in the first line denotes that the derivatives act on two different objects. Note that the symmetric part of (3.23) together with the linear constraint (3.25) implies that At this stage we would once again like to emphasise the difference to DF T W ZW . There, the background vielbein would be described by the Maurer-Cartan form of the gauge group, which

Comparison to standard and gauged EFT
Let us reflect and compare the situation here to the usual formulation of exceptional field theory.
This discussion is very similar to that in DF T W ZW , see [47][48][49], although our vielbeine are not necessarily Maurer-Cartan forms. Our 10-manifold has a GL(5) + -structure, which, when it is not locally flat, introduces a manifest background dependence through the vielbeine E ab . This is captured by the coefficients of anholonomy of the derivatives (3.23), introduces a gauging in the generalised Lie derivative (3.19) and is identified with the background embedding tensor. Closure of the algebra of generalised diffeomorphisms further requires a "section condition" (3.29). The theory thus resembles an expansion around an EFT background, as in the "gauged EFT" setup [56,58].
In the gauged EFT setup, just as in gauged DFT, the embedding tensor is determined in terms of some GL(5) + "twist matrices" where |U |= 1 and ρ is a scalar density. The precise relationship is given by the generalised Lie

34)
L 0 here has the same form as (3.20) but with τ ab,c d = τ ab = 0, W ab has weight w = 1 5 and In terms of the irreducible representations this gives We see that (3.23), which can be rewritten in the more suggestive form is similar in spirit, but there the background vielbeine E ab do not have to be GL(5) + -valued, and we use the conventional Lie derivative, not the generalised Lie derivative. Nonetheless, in gauged EFT one also finds that the section condition can be relaxed, see for example the analogous discussion for gauged DFT [61][62][63][64] and also the review [65]: for closure of the algebra one must impose the quadratic constraint on the embedding tensor, which by (3.34) automatically satisfies the linear constraint, and the section condition (3.29), where in this case the background vielbeine would be W ab , (3.33). However, one also imposes the section condition between the background and fluctuations, when acting on any fluctuations. Thus, the curved EFT formulation looks similar to gauged EFT when we have a locally flat GL(5) + -structure, as in eq. (3.3). However, even in the locally flat case, there is the difference that the embedding tensor would still be given by (3.23) rather than (3.34), and that we do not need to impose (3.38). A straightforward calculation shows, however, that in the locally flat case, where we can write In the following sections we will show that even when the GL(5) + -structure is not locally flat and we do not impose (3.38), we can use the GL(10) vielbeine E ab to construct a curved EFT formulation reminiscent of gauged EFT. However, the fact that the "background" is described by a GL(10) object while the fluctuations are in GL(5) + means that the theory is not backgroundindependent, see for example the discussion in section 5 of [66]. In contrast the usual double field theory formulation, which we wish to interpret as the "locally flat" case, has recently been confirmed to be background-independent [66] and it is reasonable to expect the same to be true of exceptional field theory. Nonetheless, the formulation presented here is manifestly coordinate invariant, and has a clear patching prescription which does not require the section condition. It can thus describe nongeometric backgrounds [51]. The interested reader can find a review of the patching discussion in double field theory in [40]. Finally, one may hope that it captures other effects, such as non-Abelian T-duality [67].

GL(10) covariant derivative
Following [47], we define a spin-connection for a vector V such that we can rewrite the generalised Lie derivative (3.20) as We then find Note that this is traceless and that for a scalar of weight w, we have so that there is no ambiguity as to whether we should be using ∇ ab or D ab for the weight-term.
It is easy to check that is a sl(5)-valued one-form and it induces a connection for GL(10)-diffeomorphisms, Γ MN P , via the vielbein postulate It is easy to check that ∇ M defined in (3.48) is a connection if ω ab,c d is a GL(10)-scalar. This follows from the tensorial definitions (3.23). As discussed in subsection 3.1.3, the connection here is a SL(5)-connection.
Finally, using (3.43) we obtain the explicit expression for the components of the GL(10)connection

Curvature, torsion and integration by parts
Let us calculate the usual GL(10) curvature and torsion of this connection. The curvature is best calculated in terms of the spin connection ω M,a b . It is given by One can check that this is still traceless so that (R MN ) a b = R MN,a b is a sl(5) element. Using the vielbeine E ab M we see that the curvature tensor lives in the To evaluate the curvature tensor it helps to note that and But from (3.23) and (3.35) we have that The rest is a tedious but straightforward calculation which shows that none of the irreducible representations (3.51) vanish, even using the quadratic constraints. We summarise the irreducible representations in appendix B.
The torsion of the connection is given by We see that for a general background, the torsion of this connection does not vanish. Let us consider its trace where in the final step we used the relation (3.31). This is important since it measures the obstruction to integrating by parts: when integrating by parts we will pick up terms such as where V M will be a diffeomorphism-density. Thus, To integrate by parts we require I to be a boundary term, which only occurs when τ ab = 0. This is consistent with the fact that supergravities with a trombone gauging do not admit an action principle. Instead they are defined only at the level of the equations of motion [68]. From the gauged supergravity perspective, this makes sense because the trombone gauges an on-shell symmetry. Indeed, in the usual gauged EFT formulation, one also finds that the trombone is the obstruction to integration by parts by a similar argument to that presented here [69].
To conclude this section, let us note that there are trivial gauge parameters, with respect to which the Lie derivative vanishes. These are given by where B a is any element of E * 5 of weight 2 5 . This is the generalisation of an "exact form" as given by the generalised Cartan Calculus [52,53] that we will discuss in the following section.

Tensor Hierarchy
In the full EFT, the fields which are "off-diagonal" between the internal extended space and the external seven-dimensional space are described by a hierarchy of tensor fields. These are related to the tensor hierarchy of maximal gauged SUGRA [54,55]. Their structure can be nicely described in terms of a certain chain complex [52,53,70]. In section 4.1 we first generalise the formulation of this chain complex [53] to take into account the curvature of the GL(5) + -structure. We then show in subsection 4.2 how this can be used to describe the tensor hierarchy. Finally we derive the topological term of the Lagrangian in subsection 4.3.

Curved Cartan Calculus
We begin by constructing the curved version of the generalised Cartan Calculus [52,53]. We want to introduce a nilpotent derivative so that we obtain a chain complex between the modules required for the tensor hierarchy, summarised in table 1. Table 1: Modules, gauge fields and field strengths relevant for the tensor hierarchy and their representations under SL(5) and GL (10). w denotes their weight under generalised Lie derivatives while λ denotes their weight under GL(10) diffeomorphisms.

Module(w,λ) Representations Gauge field Field strength
In order to avoid clutter we will from here onwards drop the λ value when referring to the modules in table 1, with the GL(10)-values always to be taken as in table 1. We will also make use of a scalar density S(1, 1) which has weight 1 under both the generalised Lie derivative and GL(10)-diffeomorphisms, but again we will refer to it simply as S (1). We also define a bilinear product • between certain modules, which maps as follows.
Finally, we want the nilpotent derivative∂ and the product • to obey the following identity [52,53]: for all Λ ∈ A(1/5) and T ∈ B(2/5) or C(3/5), We use the same • product as in the "flat case" [53], defined as and which is defined to be symmetric when acting on different modules. However, we modify the derivative∂ to bê where B ∈ B(2/5), C ∈ C(3/5) and D ∈ D(4/5). Note that these definitions also map the GL(10) weights as required, see table 1.
The derivative ∇ ab is as in (3.43) and it is important to note that∂ thus satisfies integration by parts when τ ab = 0. Let us now check the nilpotency, starting with We can split this expression into terms quadratic in the embedding tensor components, those linear in the embedding tensor components and those without. For those without we find We use the identity and the section condition (3.29) to write this as It is now easy to check using the coefficients of anholonomy thus showing that the derivative∂ is nilpotent. One can also check that this nilpotent derivativê ∂ is covariant under generalised Lie derivatives in the sense that the following diagram commutes:

Tensor Hierarchy
We now construct the tensor hierarchy [54,55] as in EFT [17] by introducing field strengths of the various potentials in table 1. Mutatis mutandis, the construction in this section is formally identical to that presented in [53]. That is, the arguments and formulae in [52,53] hold, subject to the modification of the generalised Lie derivative (3.19) and the nilpotent derivative (4.4). Thus, we will keep the discussion here brief and refer the interested readers to the original construction in E 6 [17] as well as [52,53].
The fields of the tensor hierarchy are forms of the external spacetime as well as forms of the extended space, i.e. of the chain complex (4.1). Because they can depend on both the external spacetime and the extended space, they will transform under generalised diffeomorphisms, GL(10)-diffeomorphisms and external diffeomorphisms. To account for these different symmetries, we introduce a covariant derivative for the external directions [17] (4.10) Its commutator defines a field strength where is the antisymmetrisation of the generalised Lie derivative. Although (4.11) is manifestly invariant under generalised Lie derivatives, the naive field strength F µν as defined in (4.12) is not. The deviation from covariance is however a term that generates a trivial generalised Lie derivative, i.e. it is of the form ∂ B µν ab . This intertwining between forms of different degrees is a defining feature of the tensor hierarchy, which continues by defining a field strength for B µν,a etc.
Subject to the modifications of the generalised Lie derivative and the nilpotent operator∂, we can proceed with formally equivalent definitions as for the "flat" case [53]. In particular, we define the covariant field strengths (we now drop the SL(5) indices to avoid clutter) From these definitions, one can see that the field strengths satisfy the Bianchi identities (4.14) Varying the gauge potentials leads to the following variations of the field strengths where we defined the "covariant" gauge field variations Finally, the field strengths are invariant under the gauge transformations given by (4.17)

Topological Term
We now wish to construct the analogue of the topological term of EFT [26] which reduces in the locally flat case to the topological term of seven-dimensional maximal gauged SUGRA [71].
Using the formalism described above, we construct it as a boundary term in eight external and ten extended dimensions. The proposed term is (4.18) Here we abuse notation by labelling the eight-dimensional space and the seven-dimensional external space that is its boundary by the same indices, i.e. µ = 1, . . . , 8 above. It is easy to check that the integrand has the appropriate weight under generalised diffeomorphisms, GL(10)diffeomorphisms and external diffeomorphisms. We will show that when the trombone vanishes, the variation of (4.18) is a boundary term, because this is sufficient for calculating the action.
We use the fact that when τ ab = 0 we can integrate the nilpotent derivative∂ by parts, to obtain As noted earlier, when the trombone is non-vanishing, there is no action principle because we cannot integrate by parts, mirroring the behaviour in gauged SUGRA [68] and in "gauged" EFT [69].

The action
We now wish to write an EFT action, with a curved GL(5) + -structure. This has a similar form to seven-dimensional maximal gauged supergravity, with an "external" seven-dimensional metric g µν with vielbein eμ µ . Under generalised diffeomorphisms, this external vielbein transforms as a scalar of weight 1/5, i.e.
In addition there are 14 scalars parameterising the coset space SL(5) SO (5) . We can write these in terms of the generalised metric M ab =Ẽā aẼbb δāb , where a transforms under SL(5) andā transforms under SO (5). Note that the structure group can always be reduced to its maximal compact subgroup, thus in this case from GL(5) + to SO (5), so that the existence of M ab does not impose further restrictions on the 10-dimensional manifold.
Finally there are also the field strengths of the tensor hierarchy, which have been described in detail in the preceding section 4. Schematically, the action takes the form Here, we have • S EH is an Einstein-Hilbert-like term, involving the D µ derivative, which is thus invariant under generalised diffeomorphisms, • S SK is the kinetic term for the scalars M ab , • S GK contains the kinetic terms for the gauge fields of the tensor hierarchy, • S top is the topological term, see 4.3, • S pot is the potential term, written completely in terms of g µν and M ab .
Apart from the potential, the various terms appearing in the action (5.3) are very similar to the usual EFT construction, see for example the original discussion in [17] and the specific example of SL(5) in [26], and so we will keep their discussion brief. Each term is manifestly invariant under generalised diffeomorphisms and GL(10)-diffeomorphisms, but not under external diffeomorphisms, which act as follows Here ξ µ (x, Y ) can depend on both the external and the extended coordinates. This is why we use D µ , the covariant external derivative introduced in section 4. For example D µ acts on g µν as D µ g νρ = ∂ µ g νρ − L Aµ g νρ . (5.5) The variations (5.4) are the GL(10)-covariant generalisation of [17].
We further take E ab to be independent of the external coordinates, x µ , so that It follows that The variation of E ab M also vanishes Requiring (on-shell) invariance under the external diffeomorphisms fixes the relative coefficients between the terms appearing in (5.3). We leave the details of the calculation to the appendix D.

Covariant Einstein-Hilbert term
Here we follow [25] in constructing an Einstein-Hilbert term for the external metric g µν that is invariant under generalised diffeomorphisms. The alternative is to use the vielbein formalism [17].
We can define a Riemann tensor that is covariant under external diffeomorphisms, generalised diffeomorphisms and GL(10)-diffeomorphisms as in the usual way, but everywhere replacing The Einstein-Hilbert term is then For the variation under external diffeomorphisms, see appendix D, it useful to integrate all terms involving second-order derivatives by parts to obtain -up to boundary terms - (5.12)

Kinetic terms
The kinetic term for the scalar is again defined simply by replacing the usual partial derivative with a covariant derivative ∂ µ → D µ , so that The coefficient 1 4 is required to ensure invariance under external diffeomorphisms. For the gauge potentials we use the field strengths defined in section 4.2 but we only introduce kinetic terms for F µν and H µνρ as the higher forms can be dualised to just these two. We obtain the action where the factors − 1 8 and 2 3 are required to ensure invariance under external diffeomorphisms. At this point we should also highlight that the equation of motion coming from varying C µνρ here and in the topological term (4.18) gives rise to a duality relation which in the locally flat case reduces to the M-theory duality between three-form and six-form and to the IIB self-duality. It 15) and is required for the action to be invariant under external diffeomorphisms, see appendix D.

Scalar potential
We next consider the scalar potential. This is expressed in terms of the scalar degrees of freedom which are encapsulated in the generalised metric M ab . We will calculate the scalar potential by requiring it to be invariant under generalised diffeomorphisms as well as reducing to the right supergravity action in the locally flat case and when the section condition is solved. This implies that the potential is made of two independently invariant parts where V 1 depends only on M and∇ is a connection under the generalised Lie derivative (3.20) and defined as∇ for a vector V c of weight w. We will use a Weitzenböck-like connectioñ and derive V 1 in terms of the generalised torsion of this connection, mirroring the construction in [58,64]. 6 The generalised torsion is defined as where T ab is an irrep of the generalised torsion. Explicitly, we find Explicitly, these are given bỹ and live in the 15 ⊕ 40 ′ ⊕ 10, just as the embedding tensor does [71]. We can now construct six independent generalised diffeomorphism scalar densities This form of the scalar potential is manifestly SO(5) invariant but no longer manifestly invariant under generalised diffeomorphisms. The appearance of the connection components ω ab,c d is similar to the structure of the scalar potential of DF T W ZW [47,48]. It would also be interesting to see this potential derived using a torsion-free connection, for example following [17,70,72] for the fluctuations. Before moving on, let us give the action in a form where the boundary terms have been integrated by parts (in the case when τ ab = 0 vanishes) We can also rewrite this form of the scalar potential in terms of the "big generalised metric" . This allows one to compare the scalar potential to that found in the flat SL (5) EFT [26]. The result is The details of this calculation can be found in appendix C.2. The first two lines in (5.27) reduce immediately to the SL(5) EFT action of [26] when the covariant derivatives are replaced by partial derivatives, while the final line represents a nonminimal modification in the case of fluxes, just as in DF T W ZW [47][48][49]. It is thus easy to see that when the fluxes vanish we reproduce the usual SL(5) action [26]. When they do not vanish but the GL(5) + -structure is locally flat, we expect to obtain the gauged SL(5) EFT action.

Discussion
In this paper we have shown that it is possible to define a full SL(5) EFT, on any 10-manifold with GL(5) + -structure. When the GL(5) + -structure is locally flat the formulation here reduces to the usual EFT formulation. Furthermore the action given here reduces exactly to the one found in the usual EFT formulation [26]. The benefit of the approach here is that since we are patching the EFT not just with generalised diffeomorphisms, but with ordinary GL(10)diffeomorphisms, we can also describe non-geometric backgrounds, as first discussed in [51]. In the set-up presented here usual geometric backgrounds should then be related to 10-manifolds whose structure group can be further reduced from GL(5) + to G geom , its geometric subgroup. When this is not possible, the background would be non-geometric.
One may also wonder what the physical significance of the curvature of the GL(5) + -structure is. At this point we can only speculate that this may allow us to describe non-Abelian Tdualities [67]. Also, as argued in [66], when the GL(5) + -structure is not locally flat, the theory is not background dependent. This is due to the fact that fluctuations about the background are described by elements of GL(5) + but the background, encapsulated in the GL(10) vielbeine This may suggest that we should limit ourselves to 10-manifolds with locally flat GL(5) +structure. However, as discussed in [73] for the O(D, D) case, this is a very restrictive requirement for the extended space. One may wonder what three-or four-dimensional manifolds can be described as solutions of the section condition on this restricted set of 10-manifolds.
An interesting contrast to the generalised geometry picture then emerges: as noted in [73] while for DFT / EFT, there would be restrictions on the allowed extended manifold, it is always possible to define a generalised geometry with flat O(D, D)-or E d(d) -structure on the generalised tangent bundle of any manifold. This may be reconcilable since the local flatness restriction applies to the full doubled space not to the physical manifold obtained after applying the section condition. Indeed, it has been observed in [66] that the perturbations around WZW-backgrounds can be adequately described by DFT. This seems to suggest that these manifolds admit a locally

A Closure constraints
We begin with (3.19), where τ bc,d a is for now an unspecified constant, i.e.
where Λ has weight 1 5 . The algebra closes if To keep this tractable we will call Each of these expressions will involve terms which are independent of τ ab,c d , linear in τ ab,c d and quadratic in τ ab,c d . We will denote these with the subscripts 0, 1 and 2. We find for the τ ab,c d -independent terms To simplify these expressions, note that Hence we can write Using these identities we find that the difference C a is given by Let us now turn to the terms linear in τ ab,c d , given by For later on, it will be convenient to further split C a (A.10) Finally, the terms quadratic in τ ab,c d are given by Let us begin with the quadratic terms. Their difference is given by τ bc,g a τ de,f g − τ de,g a τ bc,f g − τ dg,f a τ bc,e g + τ bg,f a τ de,c g − 1 5 τ de,f a τ bc + 1 5 τ bc,f a τ de − (1 ↔ 2) The second line can also be shown to vanish using the quadratic constraint, if we also impose the linear constraint (3.28). To see this, note that the second line can be written as with ∆ ab,cd ∈ 45. Thus we can also write it as which vanishes by the quadratic constraint (A.13). Thus we have C a 2 = 0. Let us now look at the D bc V a terms in C a 0 and C a 1 . For these to vanish we find (A.16) Thus we are lead to impose This also ensures that the terms involving V a without derivatives cancel up to the term of C a 0 , which will have to be cancelled by the remaining terms. The remaining cancellations are ensured by the linear constraint, which implies that only the 15, 40 ′ and 10 ⊂ 10 ⊗ 24 are non-zero. Thus, we can write Let us now show that with these representations C a vanishes, up to the constraint (3.30) and the section condition (3.29). We will do so by considering the individual representations in turn.
Let us begin with the 15, so that we will for now set Then we find for C a In the first equality, the first line comes from the terms of C a 1 where the free index a is on Λ 1 DΛ ac 2 and Λ ac 1 DΛ 2 , while the second and third line come from D a 1 | 15 . The totally antisymmetric terms such as D [ab ⊗ D cd] cannot contribute to the 15 so we only need to consider the commutator terms of C a 0 . They are given by We see that C a | 15 = 0 as required.
Let us now turn to the 40 ′ . For that write We start with D a 1 | 40 ′ . Note that where A def is totally antisymmetric in its indices. This is exactly the form of the terms in D 1 a | 40 ′ so we see that The remaining terms of C a 1 | 40 ′ are in E a 1 | 40 ′ and are given by On the other hand, the commutator terms in C a 0 | 40 ′ give where we use, amongst other things, that Z [ab,c] = 0. Thus, we find Putting all this together we are left with For this to vanish we require Z gh,a D gh + f a (10) = 0 , where f a (10) denotes a function of the 10-dimensional representation, which is valued in the 5 irrep.
We will now show that when this takes the form (3.30) C a vanishes by studying the 10. and determine α.
Let us, however, begin again with D a 1 | 10 . Also, we will again make use of Then, we find On the other hand from C a 0 | 10 we obtain Putting all this together we find We want to combine these different terms into expressions involving contractions of two ǫ abcde symbols, to make contact with (A.30). We find Thus, we have in total which vanishes if we impose α = −2 as in (3.29)

B Curvature
The curvature, ǫ adef g Z de,b Z f g,c .
(B.7) Then it is given bỹ The 105 gives rise to where P 480 denotes the projector onto the 480 and the . . . refer to contractions.
C Scalar potential

C.1 Lorentz-invariance vs diffeomorphism invariance
Here we will show the details that allow one to rewrite the action (5.24) in terms of the generalised metric as in (C.5).
Let us first of all rewrite the terms in (5.24) in terms of the fluctuation connectionΓ. which is analogous to the statement that the usual Weitzenböck connection is flat. Using (C.4) we can write (C.3) as in (C.2).
One can now check that it is possible to write the potential (5.24) in terms of (C.2) as follows: Let us now expand ∆ in terms of pieces independent of ω, labelled ∆ 0 , those linear in ω, labelled ∆ 1 , and those quadratic in ω, labelled ∆ 2 . Similarly, δ has terms linear in τ and ω, labelled δ 1 , and terms quadratic in ω, labelled δ 2 . We find that ∆ 0 is given by and vanishes by the section condition. This is exactly the piece which is required in the gauged EFT set-up to rewrite the action as a function of the embedding tensor in terms of the generalised metric [58].
The terms in ∆ 1 and δ 1 are not SO(5)-invariant and thus they need to cancel. Using the linear constraint we find indeed that for vanishing background trombone τ ab = 0 their contributions cancel, specifically To begin let us split the variation of the gauge field δ ξ A µ into δ 0 ξ A µ , which does not depend on M ab , and δ 1 ξ A µ , which does. Thus, the relation holds. Now the δ 0 ξ variation of the field strengths is given by where L ξ is exterior Lie derivative which is the usual 7-dimensional Lie derivative but with the covariant derivative D µ . We will also need Using these results, we find the anomalous variation of the topological term to take a simple form. Here we express it as a variation of the 7-dimensional Lagrangian, 2M ac M bd ∇ cd ξ ν g µ1ν J µ2µ3µ4,c J µ5µ6µ6,d + ∇ ab (ξ ν J νµ1...µ3 a ) J µ4...µ7 b . (D.5) Let us now turn to the gauge-kinetic term S GK . It is easy to see that the δ 0 variation of the F 2 term cancels the δ 1 variation of the H 2 term when the relative coefficients are exactly 2 3 . The δ 1 variation of F 2 will be used to cancel against the variation of the other terms and we will return to this later. For now, let us consider the δ 0 variation of H 2 . It is given by As usual [17], we take this projected duality equation to hold outside the derivative too, so that It is now easy to see that the variations (D.6) and (D.5) cancel.

D.2 Scalar potential, Einstein-Hilbert and scalar kinetic terms
Since this calculation is similar to that for the usual EFTs, see for example the original discussion [17], we will keep this section brief and mainly state results. However, we wish to emphasise again that because we will require integration by parts, the invariance under external diffeomorphisms only holds when the background trombone vanishes, i.e. τ ab = 0. For the scalar potential, we calculate the anomalous variation under external diffeomorphisms We also need the variation of the Einstein-Hilbert term (5.12). It is given by ∆ ξ R = 1 2 M ac M bd ∇ ab ξ λ ∇ cd D λ ln g − g µν ∇ cd D ν g µλ − 1 2 ∇ cd g νρ D λ g νρ − 1 2 g µν D ν ln g∇ cd g µλ − ∇ cd g µλ D ν g µν − 1 2 g µν g ρσ D µ g σλ F ρν ab ∇ ab ξ λ − 1 2 g µν F ρµ ab ∇ ab D ν ξ ρ . (D.11) Finally, we will need the variation of the scalar kinetic term. We integrate it by parts so that (D.12) It is now a simple calculation to check that the anomalous variations of the different terms cancel.