Exotic branes in Exceptional Field Theory: the SL(5) duality group

We study how exotic branes, i.e. branes whose tensions are proportional to $g_s^{-\alpha}$, with $\alpha>2$, are realised in Exceptional Field Theory (EFT). The generalised torsion of the Weitzenb\"ock connection of the $\operatorname{SL}(5)$ EFT which, in the language of gauged supergravity describes the embedding tensor, is shown to classify the exotic branes whose magnetic fluxes can fit into four internal dimensions. By analysing the weight diagrams of the corresponding representations of $\operatorname{SL}(5)$ we determine the U-duality orbits relating geometric and non-geometric fluxes. As a further application of the formalism we consider the Kaluza-Klein monopole of 11D supergravity and rotate it into the exotic $6^{(3,1)}$-brane.


Introduction Introduction
In the seminal paper of Cremmer and Julia it was shown that D = 11 supergravity compactified on a torus T n results in a d = (11 − n)-dimensional theory that exhibits a hidden symmetry described by the exceptional series E n(n) (the split real subgroup of the complexified group) [1,2,3].
Subsequent to this [4,5,6] suggested the idea that the compactified d-dimensional theory can be reformulated to make T-dualities manifest. Then a series of works suggested [7,8,9] that one could make U-dualities manifest as a exceptional field theory (EFT) living on a larger space where the U-duality symmetries appear linearly realised, i.e. as symmetries acting on the coordinates of the extended space that transforms under a particular representation R n of E n(n) . Such a space is parametrized by dim R n coordinates, n of which correspond to the usual Riemannian coordinates of spacetime, whilst the other dim R n − n coordinates correspond to wrapping modes of the extended objects of M-theory, i.e. M2-, M5-branes, KK-monopole etc. This coordinate representation R n differs for each group E n(n) and depends on the field content of the resulting theory [10,11]. Table 1   , along with their maximal compact subgroups K n , coordinate representation R n and contributing windings of extended objects.
The exotic objects will be discussed in more detail below.
A simpler case comes from compactifying D = 10 type II supergravity on a d-dimensional torus, where the duality group is just the O(d, d) of T-duality. Then the dimension of the extended space becomes doubled R O(d,d) = 2d and the extended coordinates comes from string winding modes. This is what is known as Double Field Theory (DFT) [12,13,14,15]. In both DFT and EFT, the geometry of the extended space becomes closely tied to the duality group through the local symmetries of the theory and the EFT action is invariant under the so-called generalised Lie derivative which is described in terms of the projector P to the adjoint representation of the duality group as is constructed from group invariants as described in [16,11] and further satisfy . See [17] for a general description of tensors for any Kac-Moody group.
In this paper we focus on the case of SL(5) duality group, where the coordinates transforming under the 10 of SL(5) are denoted asŶ [mn] with m, n = 1, . . . , 5 labelling the fundamental representation. To gain some intuition for the meaning of these coordinates, one decomposes them under SL(4) and identifieŝ where the six coordinates y cd describe membrane winding modes and the four coordinates x a are the usual spacetime coordinates. The section condition simply reads ǫ mnklp ∂ mn ⊗ ∂ kl = 0. (1.5) In Section 4, the geometry of the SL(5) theory is described in greater detail. The central object in this construction is the so-called generalised metric M M N which contains all the scalar degrees of freedom and is an element of the coset space SL(5)/ SO(5) (or more generally, of E n(n) /K n where K n is the maximal compact subgroup of E n(n) ).
In the series of papers [16,18,19,20,21,22,23,24,25,26] it was shown that one may consider full (supersymmetric) d + dim R n -dimensional theories that descend to the maximal D = 11, half-maximal D = 10 Type IIA/B or ungauged d-dimensional supergravity theories. In addition, upon a generalised Scherk-Schwarz reduction, these reproduce gauged d-dimensional supergravity with all gaugings expressed in terms of the generalised vielbein. The gaugings are controlled by the so called embedding tensor whose origin in EFT is the generalised torsion of the generalised Weitzenböck connection [27,28,29,30]. The role of the Weitzenböck connection and the generalised torsion in exceptional field theories was first discussed in [16] and later for DFT in [31].
From the perspective of the lower dimensional gauged supergravity, all components of the embedding tensor may be divided into two groups: geometric and non-geometric. The former corresponds to fluxes which can be turned on when compactifying from 11 dimensions. By contrast, whilst non-geometric gaugings cannot be obtained via the usual compactification of 11D supergravity, they have nevertheless been shown to descend from compactifications of EFT. A good example is given in [32] where the non-geometric gaugings of half-maximal gauged supergravity in d = 7 were obtained from a particular choice of the generalised Scherk-Schwarz twist matrix. More information on the description of non-geometric fluxes in terms of extended geometries may be found in the works [33,34,35]. For model-building applications one may refer to [36].
Just as one may consider a magnetic flux to be sourced by a monopole in electromagnetism, one may regard non-geometric fluxes as being sourced by exotic objects. Here, by an 'exotic' object we mean extended objects with tensions proportional to g −3 s or lower. As described in [37] they will have codimension two in the conventional supergravity/string theory description. They can be obtained by U-duality transformations of smeared conventional D-branes [38,39] and have tensions proportional to g −3 s or lower and are thus more non-perturbative than convention D-and NS-branes. These exotic branes indeed have also been found in string and M-theory by analyzing the spectrum of massive states in D = 3 supergravity, which fall into classifications of the E 8(8) U-duality group [37].
For each of these exotic branes, and associated non-geometric flux, one may introduce a corresponding gauge potential of mixed symmetry 1 , generically of the form A m 1 ···mn,n 1 ···np,k 1 ···kq,··· , that solves the appropriate Bianchi identity. The D = 11 types of mixed symmetry potentials contributing to the various BPS-branes were analysed in [40,41,42,43]. For each EFT with group E n(n) (n ≤ 8), one finds only a finite number of exotic branes, namely those whose fluxes can be fit into the n-dimensional internal space. A more detailed description of exotic branes and their formulation in terms of extended geometry is presented in the next section.
In terms of the conventional supergravity description, exotic branes generate backgrounds with non-trivial monodromies [37]. This means that they are not globally well defined solutions in supergravity and one needs to view the supergravity as being em-1 Throughout this paper we shall use the convention that we denote irreducible mixed-symmetry tensors of GL(D) by comma-separated sets of antisymmetric indices of non-increasing length e.g. A m1m2m3,n1n2 has the property of being antisymmetric in both pairs of indices: [n1n2] . Moreover, their irreducibility implies that taking the antisymmetric combination of any complete set of antisymmetric indices together with one index from the next set yields zero. In the example above, this means that A [m1m2m3,n1]n2 = 0. We shall often abbreviate the tensor by the lengths of the sets of indices, e.g. A 3,2 will denote a tensor of index structure A m1m2m3,n1n2 . bedded in a larger theory where the duality group is used to patch together solutions via duality transformations.
Using U-duality transformations to generate exotic branes from the conventional branes requires a sufficient number of isometries and this, in turn, implies that the brane solutions are "smeared". Furthermore, the brane solutions have low co-dimension and the harmonic function describing the solutions have poor behaviour at infinity (various aspects of this are discussed in [37]). In a series of works [44,45,46,47,48,49] it was shown that including the worldsheet instanton corrections to the smeared background can localise the brane, breaking the isometry and thus recovering the full non-smeared solution, albeit with fields now depending on dual coordinates. This is already true for the Kaluza-Klein monopole whose instanton-corrected version, called the localized KK-monopole, is no longer a solution of the supergravity equations of motion. Instead, as shown in [45,50], it solves the equations of motion of DFT. In the work [51] the T-duality chain has been investigated further and the backgrounds describing the so-called exotic Q-and R-branes have been obtained. When smeared along all the dual coordinates these correspond to the 5 2 2 and 5 3 2 branes of Shigemori and de Boer. Moreover, if one allows dependence on the compact winding coordinates, the Q-brane background becomes precisely the instanton corrected 5 2 2 -brane of [47]. 2 Hence, one may consider the approach taken by DFT and EFT as one which allows us to generate instanton-corrected backgrounds of exotic branes in a systematic manner.
This papers continues research in this direction and studies the exotic branes of Mtheory described in the framework of the EFT with group E 4(4) ∼ = SL (5). This corresponds to a d = 7 supergravity augmented by a 10-dimensional extended space. We start with the background of the KK6-monopole, uplifted to the full (7+10)-dimensional theory, and apply an SL(5) transformation to obtain the background of the 5 3 -brane. Previously, [54] the usual standard 1/2 BPS branes of eleven dimensional supergravity were described from an EFT perspective and shown to be given by a single EFT solution with different solutions of the section condition. In this paper, this description is extended to include these exotic branes showing how from the perspective of EFT, the exotic and non-extoic objects are all on the same footing. To put it another way, exotic branes arn't exotic from the point of view of EFT.
We will following the bottom-up approach of constructing and studying the various exceptional field theories directly for the U-duality groups. In a very influencial work, it was conjectured [55] that E 11 should provide a fundamental theory of M-theory and subsequnet to that the the various EFTs follow from this construction [56,57,58]. The particular case of SL(5) EFT was investigated in view of this in [59]. Although we will not make use of the top-down construction of the SL(5) EFT, we shall sometimes use the E 11 construction to keep track of the various potentials and U-duality orbits of the solutions, in particular in section 5. This is similar to what was done in [60,61,40,41,43]. One should note the relevant work of [62] in constructing dual fluxes in EFT and crucially the work of [63] in the construction of the the EFT SL 5 fluxes.
The paper is structured as follows. In Section 2 we give an overview of exotic branes in M-theory and their classification under a U-duality action. In this section we will use the E 7(7) EFT which provides a concise description of the expected exotic branes. In Section 3 the embedding tensor of the SL(5) theory is considered and it is explicitly shown how its components relate to the fluxes of the D = (7 + 4)-dimensional theory under the decomposition SL(5) ←֓ GL(4), and how one can transform the fluxes into each other by the action of an SL(5) transformation. Finally, in Section 4, the desired backgrounds are explicitly given whilst the corresponding fluxes are calculated in Section 4.4 to confirm the interpretation that the background is indeed sourced by the 5 3 -brane.

Exotic branes in Type II and M-theory
We start by describing how the exotic branes arise in the E 7(7) EFT. From the supergravity point of view an exotic brane is not a globally well defined codimension 2, solution since the solution describes is a monodromy around the brane that is not part of the diffeomorphisms of supergravity. But for EFT they are globally well behaved since the monodromy around the exotic brane is a subgroup of the local E 7 of the EFT.

g −3 s -branes from EFT
The generalised torsion of the E 7(7) EFT has components transforming only in the 912 and the 56 representations of E 7(7) [64,65,66,67,68,20]. These are decomposed under the embedding E 7(7) ←֓ SO(6, 6) × SL(2) as follows: (2.1) The group SL(2) is the one that transforms the axio-dilaton, and hence by further decomposing SL(2) ←֓ R + to obtain the dilaton scaling and thus a relation to the g s power, we obtain the following: The R + weight here is naturally related to the power of g s appearing in the expression for the mass for the corresponding state in 4D maximal supergravity or, equivalently, in the expression for tension of the corresponding brane (see below  Figure 1: Branes of Type II theories whose tension is proportional to g −α s with α ≤ 4 and the relations between them and the M-theory branes (note that not all branes with α = 4 are shown here). The black, red and blue arrows respectively denote reduction/oxidation, T-duality and S-duality.

SL(5)
The relation to this power of g s is obtained by an appropriate shift of the R + weight. We shall follow the conventions of [69] and identify the irreducible representation 32 2 with the RR fluxes (T ∼ g −1 s ), 220 1 with the NS-NS fluxes (T ∼ g −2 s ) and the 352 0 with the non-geometric P-fluxes (T ∼ g −3 s ). The others correspond to non-geometric fluxes sourced by branes whose tension is proportional to even lower powers of g s .
To single out the fluxes in their conventional six-dimensional tensorial form, one must further decompose the representations under the embedding of O(6, 6) ←֓ GL (6), that gives the fluxes for g −α s with α ≤ 3 3) where the conventions are chosen to be adapted to the IIA reduction and the labels are related to the scaling of the object's tension with the M-theory circle. At the level of tensors, the above decomposition reads (2.4) Note that the fluxes at the level α = 2 contain the components f a and f a , which correspond to the gaugings F M of half-maximal supergravity. Such gaugings do not correspond to fluxes sourced by BPS branes or trombone gaugings of the maximal theory. They are instead equal to the trace parts of the components F ab c and F a bc of the generalised torsion, with the remaining traceless components being the regular Q and f fluxes. The same is true for the fluxes P ab and P abcd , which shall be omitted from further discussion together with F M . For the Type IIB reduction one would obtain the RR fluxes inside the θȦ tensor and the P-fluxes inside the θ M A tensor (of opposite chirality). Alternatively, one could embed the GL(6) differently.
In D = 10 one generally finds all the fluxes corresponding to α = 3 inside the tensor θ MȦ (or θ M A ) of O(10, 10), for which we write (2.5) Here, we are using the notation common in the DFT literature and are decomposing the O(10, 10) generalised index M = 1, · · · , 20 into the two sets ( a , a ) with a = 1, · · · , 10. Since we know how T-duality acts on the irreducible representations of O(10, 10) it is straightforward to recover the T-duality transformation rules for the P-fluxes. Indeed, the first index of P x,b 1 ...bn (resp. P x b 1 ...bn ), which comes from the vector index of O(10, 10), is lowered (resp. raised) if the T-duality transformation T x acts along x. The rules for indices in the second set are the same as for the RR-fluxes, i.e. if the direction x is in this set, then T x removes the corresponding index and adds it if it is not. All of this is summarized as follows [69]: In [40,69] it was shown that the corresponding gauge potentials for the P -fluxes listed above are the following ⊕ E 9,2,1 ⊕ E 9,4,1 ⊕ E 9,6,1 ⊕ E 9,8,1 ⊕ E 10,4,2 . (2.7) To illustrate how the correspondence between fluxes work, let us consider the IIB flux P a,bc that is the S-dual of Q-flux. Since these are all magnetic fluxes, we shall first perform Hodge-dualisation in 10 dimensions and write P a,bc −→ P a 1 ...a 10 ,bc,a = 10∂ [a 1 E a 2 ...a 9 ],bc,a + · · · , (2.8) where the ellipses denote other contributions from gauge potentials of lower rank. Hence, we arrive at the IIB potential E 9,2,1 . Similarly, we can understand the fluxes with an upper vector index, say P a bc , as P a bc −→ P a 1 ...a 9 ,bc = 9∂ [a 1 E a 2 ...a 9 ],bc + . . . , (2.9) which gives the IIB potential E 8,2 . To recover the potentials of the form E 10,p,q one must consider all the wrapping rules in their full glory and the reader is referred to [40]. The wrapping rules for exotic branes tell us that all indices must take values such that one ends up with a form in the non-compact space in order to end up with a consistent compactification. In other words, if there are extra sets of indices as in E 9,5,1 , the sets of indices on the right must take values in the compact dimensions and moreover be repeated. The remaining must form worldvolume directions of the corresponding brane. Hence, say for E 9,5,1 we must write with x 1 , . . . , x 4 and y in the compact directions and a 1 , . . . , a 4 in the non-compact directions. This couples to the worldvolume of a 3-brane. To identify the brane, one notices that the counting implies that the tension of the brane coupled to E b+c+d,c+d,d is proportional to d radii to the third power, c radii to the second power and to b radii entering linearly. Hence, following the conventions of [37] we write with the subscript 3 indicating that these are g −3 s objects, whose masses in type II string theory are given by: Starting with the T-and S-duality rules for the fluxes one can determine the corresponding rules for the branes as follows: (2.13) Using these rules it is straightforward to compose Figure 1 containing all branes with α ≤ 3 and some of the branes with α = 4. The corresponding gauge potentials that these branes interact with and the co-dimensions n of the branes have been collected in Table  2 IIA IIB Co-dimension Potential Brane Potential Brane All the listed branes of Type IIA should descend from those of M-theory and so this allows one to recover the latter set, given the spectrum of exotic fluxes in Type IIA. Whilst, in principle, the exotic fluxes of M-theory could be classified as components of the corresponding embedding tensor gauging the full U-duality group there is added complexity here in that the corresponding group E 11 is not understood well enough. Hence, we shall use the following lift/reduction rules instead to conjecture new exotic branes and their corresponding gauge potentials: where, in the second line, the reduction is performed along a direction x which is chosen to be the M-theory circle. The result will differ, depending on whether this direction is chosen to be along or transverse to the worldvolume of the brane, or along its special (quadratic or cubic) circle. Using these rules we have extended Figure 1 with non-geometric branes living in 11 dimensions that give rise to all branes in Type IIA with α ≤ 3.
In this paper we are interested in the 5 3 -brane of M-theory that we shall obtain from an SL(5)-rotation inside the SL(5) EFT. To investigate the reductions of branes from 11D into Type IIA it proves extremely useful to introduce the following notation [37]: where a denotes the number of spatial longitudinal directions of the brane. For example, the 5 2 2 -brane becomes (252) 2 in this notation where the second digit denotes the number of longitudinal directions and the third denotes the number of special directions which enter quadratically in the expression for mass.
The reduction rules for the 5 3 = (253), are simply where the underlined group denotes the direction along which we compactify. Since we know the gauge potentials sourced by the 4 3 3 -brane from the table above, and that of the 5 2 2 -brane from [70], we can determine the gauge potential sourced by the 5 3 -brane as Above, we have denoted the M-theory circle by ♯ and the eleven-dimensional indices bŷ M = 0, 1, . . . , 10 that reduce to M = 0, 1, . . . , 9 and ♯. In the first line we reduce along a "quadratic" direction, meaning we choose the compact direction ♯ to be in both the first and second sets of indices, ending up with a type IIA potential of type B 8,2 . It was shown in [70] that the 5 2 2 -brane precisely interacts electrically with this potential. In the second line we reduce along a linear longitudinal direction and hence wrap the 5 3 -brane itself on the M-theory circle. Specifically, we put the direction ♯ inside the first set of indices whilst keeping the second set free and this gives us a type IIA potential E 8,3 .
Formally, we can also choose to put the M-theory circle ♯ only into the second set of indices. This does not correspond to a wrapping or reduction of the 5 3 -brane and there should not be any BPS-brane associated with the resulting type IIA potential E 9,2 [41] that formally lies at the g −3 s level [60]. Finally, if ♯ is in neither the first nor the second set of indices, we are reducing along a transverse direction and obtain the gauge potential E 9,3 . This is conjectured to be the source of the 5 3 4 -brane, in agreement with the compactification rules for the masses of the corresponding states.
In the following sections we shall show that the KK6 solution in SL(5) EFT, rotated by a certain matrix U ∈ SL(5), indeed sources the desired D = 11 gauge potential D 9,3 . The corresponding magnetic flux is the one denoted by Q a,b in [63] that appears inside the gaugings 10 and 40 of SL(5).

Brane-counting in SL(5) EFT
In the previous section, it was shown how fluxes magnetically sourced by both standard and exotic branes descend from the embedding tensor of the E 7(7) theory and how they can be organized into the tensor θ M A in the vector-spinor representation of O(10, 10). The focus above was only on the branes whose tension is proportional to g −3 s , while in general the embedding tensor of the E 7(7) theory provides many more fluxes and generates branes with tension proportional to up to g −7 s [40,41,69]. It is, however, interesting to classify the branes living in M-theory whose fluxes fit a chosen U-duality group. To retain control over the proliferation of irreducible representations, let us focus on the SL(5) exceptional field theory, whose embedding tensor lives in Θ mn,k l ∈ 10 ⊗ 24 → 10 ⊕ 15 ⊕ 40, (2.18) where m, n = 1, · · · , 5 label the fundamental representation of SL (5). The 11D SUGRA is embedded in the SL(5) EFT as SL (5) ←֓ SL(4) × R + , that induces the following decomposition To represent the actual fluxes sourced by branes that fit the internal 4 dimensions of 11 = 4 + 7 split, these representations must be combined into reducible tensorial expressions. Analysis and further comparison with branes of the 7+(3+3)-dimensional DFT show that the following combinations work and give the expected fluxes in the M-theory decomposition T a = f ab b , where we employ the notation of [63] that the letter in superscript of an irreducible representation denotes its origin and where confusion may occur and indices a = 1, · · · , 4 label the directions of the M-theory torus. The remaining 4 θ 3 coming from the trombone gauging does not correspond to a BPS brane.
As is the case for the DFT, we shall identify fluxes by the representations of the above decomposition. Then, as will be shown later, the torsion of the so-called Weitzenböck connection will be associated with the magnetic fluxes.
These fluxes may then be expressed in terms of potential fields, say F abcd = 4∂ [a C bcd] + . . . where the ellipses denote additional terms which deform the naïve first term into the full flux sitting in a U-duality irreducible representation. The precise form of the expressions depend on the frame chosen, which is a U-duality analogue of the switch between the B-field and the β-field in DFT. Hence, to escape such ambiguities, we shall always associate the irreducible representations with the fluxes rather than the potentials.
Using the conventions in the previous section, it is easy to see that the listed branes indeed couple to the obtained fluxes.
Then using the rules for reduction from 11D to the Type IIA theory in ten dimensions, one is able to obtain the spectrum of branes of Type IIA from the above spectrum. However, to compare with the branes coming from the analysis of the O(3, 3) DFT below, one should take into account that not all Type IIA branes are able to fit their associated magnetic flux into 3 internal directions.
Before presenting the reduction rules let us consider the embedding of DFT inside the EFT, that is given by SL (5) (4), the resulting decomposition is superficially the same as above. However, the gl(4) is embedded differently to above. To obtain the fluxes interacting with Type II branes one must consider the common gl(3) of the two gl(4) subalgebras. In terms of the embedding of O(3, 3) ←֓ GL(3), one obtains (2.21) The subscripts above denote the scaling under the diagonal R + ⊂ GL(3). To identify the corresponding branes, we first notice that the irreducible representations 20 3 + 4 3 of O(3, 3) can be written as a vector-spinor θ MȦ .
The resulting irreducible representations then correspond to the fluxes of g −3 s branes. Similarly, the irreducible representations 10 −2 ⊕10 −2 combine into two bispinors symmetric in the pairs of indices, θ (AB) and θ (ȦḂ) which, upon contraction with gamma matrices, form θ M N K (for details, see the lecture notes [71]). Hence, these give fluxes for the g −2 s branes. Finally, the irreducible representation 4 −7 corresponds to a DFT spinor θ A and so contains the RR-fluxes which, for our case, are sourced by D6 and D8 branes. The remaining 1 8 corresponds to the 5 3 4 -brane, whose tension is proportional to g −4 s , that is the lowest power of g s one can get in d = 7. Note that, according to [41,69], the lowest power −α of g −α s obtainable in d = 4 is α = 7, but can go as low as α = 11 in d = 3. This is because more and more fluxes are able to fit into the would-be internal directions of the corresponding EFT. Figure 2: Branes whose magnetic fluxes fit the SL(5) EFT (the first line) and whose magnetic fluxes fit the O(3, 3) DFT (the second line). The lines further below distinguish between branes whose tension is proportional to g −α s for α ≤ 4. Representations of GL(3) on the intersection of a column for a Type II brane and a row for a power of g s show how the corresponding flux transforms inside DFT Now we are ready to present the full table containing all possible (magnetic) branes fitting the SL(5) EFT and the O(3, 3) DFT, see Figure 2. As was the case for fluxes in the SL(5) theory, some of the fluxes here are not irreducible as representations of SL(3) e.g. the flux for the 5 2 2 -brane belongs to 6 2 +3 2 but, as an SL(3) tensor, this can be written as which is the usual Q-flux indeed sourced by the brane.

Orbits of the SL(5) U-duality group
The components of the generalised torsion of SL (5) EFT are irreducible representations that can be characterized by their weight diagrams. The Lie algebra sl(5) maps from the different weight spaces to one another. A subset of the sl(5) transformations are the usual infinitesimal diffeomorphisms and they do not rotate different types of fluxes as in the decomposition (2.19) into each other.
We use the following strategy to determine how the U-duality group acts on fluxes. First, one determines the simple root of a U-duality group that becomes decoupled under the embedding of G ←֓ GL(d), where G is the corresponding U-duality group and GL(d) represents internal diffeomorphisms. Next, one constructs weight diagrams for each of the irreducible representations contributing to the embedding tensor and then determines the branching into irreducible representation of GL(d). Finally, the connections between the weights of different irreducible representations of GL(d) inside a given irreducible representation of G, generated by the decoupled root, will precisely give the U-duality orbits. We shall demonstrate this procedure by applying this idea to the SL(5) EFT.

Root system of SL(5)
The simple roots of the Lie algebra sl(5) in the canonical ω-basis of fundamental weights are given by the following  In addition, one has the same number of negative roots and four Cartan generators. We shall use the simple root α 45 to be the one associated with the decomposition SL(5) → GL(4). A convenient basis to produce the appropriate SL(5) rotation matrices for generating the non-geometric backgrounds later is as follows where µ m are the basis vectors of the fundamental representation which all lie in different weight spaces. A vector in this representation is expanded as V = V m µ m . In this basis the action of a root −α mn (for some fixed m and n) will transform the vector component V m into V n . A similar notation will be employed for the conjugate co-vectors. Since the Lie algebra elements act as a derivative, the action of the root −α mn on a tensor will produce a sum of terms with the corresponding index conversion in each term. For example one can write where one notices that the dual representation transforms in the opposite way as it should. The signs on the right-hand sides will be of little relevance for our discussion.
Using the weight diagram Figure 3, it is straightforward to recover the explicit form of the matrix corresponding to SL(2) rotations generated by the subalgebra (α 45 , −α 45 ) and the corresponding Cartan generator. Explicitly, we have (5), (3.5) which rotates the vectors µ 4 and µ 5 into each other. The notation U 123 was chosen to comply with that of [63]. In their notation, one may understand the subscript numbers as corresponding to the weights orthogonal to the rotation plane. To show that the above matrix indeed represents a rotation in the plane (µ 5 , µ 4 ) we first notice that these weights transform in the fundamental representation of SL(2) under the action of the roots α 45 , −α 45 and the corresponding Cartan generator. Hence, one writes and the lower right block of the matrix U 123 is simply obtained as We shall see that the transformation that transforms KK6 into the 5 3 -brane is given by u 123 [π/2] and reads Hence, this group element simply replaces V 4 ↔ −V 5 and more generally it interchanges all indices 4 ↔ 5 on any SL(5) tensor (up to sign). As is common for elements of the form (3.7) with ϕ = π 2 the finite duality transformation is (the covering of) a Weyl transformation, namely an elementary transposition in the symmetric group S 5 .

Orbits inside the 15 representation
Let us start with the representation 15 that contains the fluxes F abcd , Q (a,b) and the geometric flux f ab a corresponding to the M5, 5 3 branes and the KK6 monopole respectively (cf. (2.20)). Note however, that the flux components f ab a are non-zero not only for the KK-monopole bacgkround, but also for the 5 3 -brane and for other solutions. This is related to the fact that we consider the full 11D solutions rather than their truncated 7D or 4D versions. The same was true for the DFT solutions in [51]. The structure of this representation is schematically depicted in Figure 4. Since the branching SL(5) → SL(4) × GL(1) is obtained upon deleting the root α 45 , it is not surprising that this root allows one to travel between different irreducible representations of SL(4) inside SL (5). In other words, the corresponding operator U 123 is precisely the desired U-duality transformation that transforms the flux F abcd into the Q-flux and the trace part of the geometric flux f ab c .
To understand the precise action of the transformation U 123 on the weights of Figure  4, one first notices that the components (Y 55 , Y 54 , Y 44 ) transform as a three-dimensional representation of the SL(2) subgroup generated by α 45 and −α 45 and the Cartan element, while the three pairs (Y i5 , Y i4 ) with i = 1, 2, 3 all transform in the fundamental representation. Hence, for the latter, one simply concludes that the transformation just rotates the weight in each pairs into each other. Hence, we have the following U-duality orbit for each of the pairs in the fundamental representation (3.9) The three-dimensional representation is more subtle and, to derive the orbit, one first writes the operators E α 45 and E −α 45 as Now, simply calculating u 123 and setting ϕ = π/2, we obtain the following matrix This implies, that the U-duality transformation U 123 for ϕ = π/2 rotates Y 55 into Y 44 , while Y 54 just acquires a prefactor. This is in agreement with the general property that U 123 is a Weyl transformation exchanging the indices 4 and 5 (up to sign). In terms of flux components one writes (4,4) , (3.12) These U-duality orbits are indeed consistent with the ones obtained in [63]. Hence, we conclude that the U-duality transformation given by the matrix U 123 for ϕ = π/2 rotates the M5-brane into the 5 3 -brane, the KK-monopole into the 5 3 -brane but leaves some component of the flux unchanged.

Orbits inside the 40 representation
The 15 component of the embedding tensor Θ mn,k l contains only the fluxes sourced by the M5, 5 3 -brane and the KK-monopole. In order to get the complete spectrum of all branes, one must consider the other embedding tensor representations as well. Hence, we now turn to the representation 40 of SL(5) whose weight diagram is depicted on the Figure 5.
The irreducible representation 40 of SL(5) has the subtlety that 10 pairs of vectors of this space have the same weights i.e. there are non-trivial weight multiplicities. This is related to the fact that the components Z ab,c can be symmetric or antisymmetric in {b, c} but still possess the same weight. Hence, depending on the index arrangement, these pairs either stay entirely in one irreducible representation or break into two vectors living in different irreducible representations under branching.   For example, the components Z 4(1,2) and Z 4 [1,2] both belong to the 20 3 , while the components Z 5(1,2) and Z 5 [1,2] descend into the 10 −2 and 6 −2 respectively. Using the fact that each root of the algebra acts as a derivative, it is straightforward to construct the diagram in Figure 5, which explicitly shows the action of the simple roots on each weight 3 . There is still a subtlety relating to the action of the positive roots corresponding to raising operators. The action of −α 23 on Z 21,2 gives Z 31,2 + Z 21,3 = −2Z 1 (3,2) , up to some prefactors. Hence, according to Figure 5, one cannot lower the weight Z 21,2 to Z 3(2,1) by the action of −α 23 . However, a straightforward check shows that raising both the weights Z 3(2,1) and Z 1 (3,2) gives just Z 21,2 . Hence, it is neccessary to keep this in mind and always keep track which plane is used for a U-duality rotation generated by a pair of roots (α, −α).
Taking this subtlety into account, we write the following for the relevant root α 45 Similar to the 15 representation, we observe both doublets and triplets here with respect to the action of the SL(2) generated by the α 45 root. In addition, one has a scalar Z 5(i,4) − Z 5[i,4] which corresponds to a self-dual combination of the T and Q flux components. Combining these rules with the rules from the previous subsection one can write the full U-duality orbit under the action of the operator U 123 for ϕ = π/2 as (3.14) Self-duality of the components T (i,4) and Q ij follows from the fact that the operator U 123 acts on the combination Z 5(i,4) + 3Z 5[i,4] of the triplet trivially, according to (3.11), whilst the linearly independent combination Z 5(i,4) − Z 5[i,4] is a scalar on its own. The same is true for the self-duality of F a4 a . Note that the components of the traceless part of the geometric f -flux are encoded inside Z ab,c . Taking this into account and redefining {x = x 1 , y = x 2 , w = x 3 , z = x 4 } one finds these U-duality orbits are in perfect agreement with what has been found in [63].
In the following section, we will focus on a field realization of the above U-duality orbits and construct the corresponding solutions in EFT. In particular we consider the KK6 monopole, which will be shown to have a vanishing Z 4(i,j) component. Hence, according to the above U-duality orbits, it will be shown to give the 5 3 -brane upon dualization.

Uplift of SUGRA solutions into the SL(5) EFT
We now consider explicit solutions of the SL(5) EFT and their duality orbits.

SL(5) theory
Let us start with the explicit form of the generators and projectors for the fundamental and two-form representation: For consistency, the weight of a generalised vector Λ mn must be set to λ(Λ) = 1/5. The generalised metric discussed below will also have weight λ(m mn ) = 1/5. Note that this convention is different from [25] which is related to a different realization of the generalised metric in terms of the fundamental fields, and the fact that it has determinant not equal to one. The overall factor of 1/2 is needed to make connection to the conventional GL(4) Lie derivative when solving the section constraint (see [59,63]). Since only the relative coefficients are fixed one is free to choose the factor in front of the projector and this, in turn, defines the remaining factors. We fix our coefficients to match common conventions in the literature [59,72,10]. To ensure consistency, we impose that such transformations close onto the following algebra with the following section constraint Decomposing the projector into Kronecker symbols one obtains the following simple expression for the transformation of a vector of weight λ: which indeed coincides with [72,10]. For the generalised metric we have As an element of R + × SL(5)/ SO(5) the generalised metric is represented as the following matrix in terms of the fundamental fields 4 Here the factor e φ = g 1 7 is given in terms of the determinant of the seven-dimensional external metric and is needed to make the generalised metric transform as a generalised tensor (see [63]). This will result in a transformation of the external 7-dimensional metric g µν under a U-duality rotation of the internal space. Alternate field parametrizations, where the SL(5) transformations do not act on the external volume, are also possible but we use the one above to conform with [25,63]. The corresponding vielbein and its inverse read where e = det eā a = h 1 2 is the square root of the determinant of the metric h ab in the internal sector. In what follows, we will also be using a different parametrization of the generalised vielbein based on a trivector Ω abc , for which we have The generalised metric for this parametrisation is then given by As was the case for the T-duality orbit of the NS5-brane in [51] it will be more natural to calculate flux components using the above parametrization. Finally, we couple the internal sector to a seven-dimensional external space (indexed by µ, ν = 1, · · · , 7) to write the full SL(5) EFT action as (4.13) where the kinetic terms are given by  14) and the potential term is as follows  In addition, the variation of the Chern-Simons term is given by (4.17) where we have corrected a misprint in the prefactor in the same formula in [25].
The uplift of a solution of 11D supergravity, written only in terms of the metric, into a thus defined EFT is then trivial. Indeed, one just starts with a 11=7+4 split of the solution and substitutes the fields into the generalised metric. The fact that the 11D supergravity is fully embedded inside the EFT guarantees that such a background is indeed a solution. Then one is free to rotate the fields by an SL(5) transformation to recover other solutions of EFT which, however, may no longer be geometric solutions of 11D supergravity.
The uplift is performed trivially by identifying the geometric coordinates {y a } of the spacetime with the subset {Ŷ a5 } of the extended coordinates, under a further SL(5) ←֓ SL(4) splitting as shown in Table 4. Table 4: The trivial lift of the KK6 background to EFT, identifying the internal space with a subset of the extended space. In this frame, the section aligns with the eleven-dimensional physical spacetime, namely (x µ ,Ŷ a5 ).
The generalised metric for the uplift of the KK6 solution is obtained by simply substituting the metric h ab into m mn in (4.8) with C (3) = 0.

5 3 -brane
To obtain the (smeared) 5 3 -brane we consider the following SL(5) rotation under which the matrix of coordinates transforms as follows: More succinctly, we represent this aŝ which corresponds to the U-duality transformation U 123 -an element of the SL(2) subgroup of SL(5) generated by the root α 45 , as discussed earlier-which should rotate the KK-monopole background into the 5 3 -brane. Later, we shall show explicitly that the KK-monopole background triggers the geometric f -flux as well as a component of the trombone gauging. According to [63], and the previous sections, the U-duality rules acting on those components of the geometric flux should give the Q-flux, when acted on by U 123 . The generalised metric after the rotation is given by In the non-geometric parametrisation (4.12), one obtains the following background where H becomes a harmonic function of only the extended internal coordinates (4.24) Before commenting on this naïvely strange result let us also write the background in the C-parametrization (4.8) where K = H 2 + A 2 . Note that the powers of the harmonic function differ from the 5 3 background of [63] since the function H here is harmonic 3-dimensional (flat) space rather than 2. Despite this, the overall structure of the solution shows the same relation to the 5 3 -brane of [63] as the Q-brane solution obtained in [51] (which depends on one non-geometric coordinate) does to the 5 2 2 -brane of Shigemori and de Boer. Table 5: Embedding of the obtained 5 3 background into the extended space. Here • denote directions on which the harmonic function depends and ⊗ denotes smeared direction. Upon reduction to the 11D space-time the three non-geometric directions will correspond to the special directions of the 5 3 -brane As mentioned above the resulting solution depends only on the extended internal coordinates that are not in the original M-theory. The reason for this is that we started with a Kaluza-Klein monopole that fills all seven external dimensions and thus has its transversal directions, on which the original solution depends, completely in the internal space. As the U-duality U 123 exchanges all the geometric coordinates y 1 , y 2 and y 3 for non-geometric coordinates we find the fully non-geometric 5 3 solution.

Generalised torsion and fluxes
Given the generalised Lie derivative L the generalised torsion may be defined in the usual way where the superscript denotes the derivative used to calculate L. To make contact with the flux formulation of EFT, one defines the covariant derivative using the so-called Weitzen-böck connection, that is where Vm m is the generalised vielbein with barred (flat) indices referring to the tangent space. Such a connection ensures that the generalised curvature tensor vanishes and that the remainder in the commutator of covariant derivatives transforms as a tensor, coinciding with the generalised torsion as defined above. Moreover, understanding the generalised vielbein as a twist matrix of a generalised Scherk-Schwarz reduction ansatz, this torsion precisely reproduces the embedding tensor of gauged supergravity, see for example [27] and references therin. Since the latter is related to internal constant fluxes in compactification schemes it is natural to understand it as a general duality-covariant definition of non-constant fluxes, or with slight abuse of terminology, field strengths. Substituting the Weitzenböck connection into the definition of the torsion, one obtains the components where we denote the flattened derivative by ∂mn := V mm V nn ∂ mn . This may be rewritten as with the irreducible components defined in the same way as in [27] 10 :

15
: (4.30) Substituting in the explicit expression for the generalised vielbein and performing the decomposition SL(5) × R + ←֓ GL(4), one may rewrite the irreducible components of the torsion in terms of the fundamental fields. As we will be mostly focusing on only the KK-monopole and the 5 3 -brane, we shall not present the general expression here. The interested reader may find them in [63].
Let us now turn to the explicit backgrounds, starting with the KK-monopole. The only non-vanishing components read (4.31) Hence, we see that only the geometric flux f ab c along with its trace is turned on-precisely the fluxes associated to the KK-monopole.
To avoid confusion, it is worth noting that it is not surprising to see the trombone gauging turned on. Indeed, the same observation has been made in [51] where all the obtained backgrounds, including the KK-monopole, have been shown to trigger the FM gaugings of DFT. Similar to the trombone gauging, these should always be set to zero in consistent truncations on compact manifolds. Since we are considering the full theory in 7+(4+6) dimensions, the background obtained here is clearly not a consistent truncation into either 7 or 4 dimensions and the non-zero trombone gauging of the generalised torsion is a perfect indication of this. We thus see that using the word "gauging" for components of the generalised torsion is somewhat misleading and is used here only for the sake of succinctness and to avoid usage of the even more confusing "field strength". Now we turn to the 5 3 -brane background, for which we find the following non-zero components of the generalised torsion: where we Hodge-dualized the lower index a and introduced hatted indicesM = 0, · · · , 10 to label the eleven coordinates. The main contribution to the field strength F 10,3 dual to the embedding tensor Q 10,3 comes from the gauge potential D 9,3 according to Hence, we conclude that the brane corresponding to the obtained background should electrically couple to the gauge potential D 9,3 . By simple counting, one may check that the corresponding brane is 5-dimensional. Indeed, since the upper indices must always be contracted with the lower ones according to the wrapping rules, we are left with only 6 indices for the gauge potential, which correspond to 6D worldvolume of a 5-dimensional object. The final step is to check what gauge potentials one recovers upon reduction from 11D to the Type IIA theory along a circle coordinate denoted ♯. We split the index aŝ M = (M, ♯), with M = 0, . . . 9, which gives the following components: where we have used the wrapping rule of [40,73,74,75] that states that directions in the upper set should be always repeated in the lower set, but not the opposite. Hence, one may put ♯ in the upper set (and thus necessarily in the lower set as well) to obtain the gauge potential B 8,2 . Alternatively, one can choose ♯ to be in the lower set, whilst keeping the upper set free of ♯, to obtain the potential E 8,3 . The former corresponds to the 5 2 2 -brane, whilst the latter gives the 4 3 3 -brane-in perfect agreement with Figure 1. Finally, all the indices of the gauge potential D can be chosen to be transverse to the M-theory circle, which gives the gauge potential E (9,3) in Type IIA, that corresponds to the 5 3 4 -brane again in agreement with the rules for reduction of the masses of states in 2D supergravity. Note that this is an α = −4 brane and hence is absent in the classification of Riccioni et al.
All of this allows us to conclude that the obtained background indeed corresponds to the 5 3 -brane of M-theory.

E 11 embedding and U-duality orbits
In this section, we revisit the U-duality transformations of the previous section by using an alternate identification of the solutions using the roots of E 11 . The conventions we are using for labelling these roots is the same as in [55,60]. In particular, the exceptional root of E 11 will be labelled by root number 11 and its associated simple Weyl reflection w 11 implements a double T-duality in the last two directions 8 and 9 of the ten-dimensional space-time, whilst also interchanging them. The S-duality node has number 10 and its associated Weyl reflection w 10 implements S-duality.
As in previous work on solutions and E 11 [76,77,78,79,80,81,61], all solutions couple electrically to their corresponding potentials e.g. the M5-brane couples to a sixform potential. The potentials and U-duality orbits can then be studied in an appropriate decomposition E 11 → E 11−D × GL(D) [66,67]. Moreover, using the identification of generalised space-time with the fundamental representation of E 11 , often called ℓ 1 in the literature [82,83], we can also perform the U-duality transformations directly on the coordinates.
Recall that the coordinatesŶ mn transform in the 10 (with derivatives in the 10) of SL(5) and they decompose them in the M-theory 4 solution of the section condition as We identify the directions y a (for a = 1, 2, 3, 4) with the directions x ♯ , x 9 , x 8 , x 7 (in that order) in the tables below. F 55 .

From M5 to 5 3
Let us consider the non-gravitational duality chain M5 →NS5→ 5 2 2 → 5 3 . This can be arranged for example as follows (see also [63,Sec. 5.3]  One can follow the masses through the various steps using the rules (2.13) and (2.14) to verify the various objects. The above is classically thought of as a smeared M5 brane with a harmonic function depending on x 1 and x 2 everywhere, more in the sense of a 2D U-duality multiplet [81].
Let us first try to understand this process. The original smeared M5 brane has a field strength component This component is a singlet of SL(4) and of SL (7). (There are also other field strength components, but they are not SL(4) singlets nor SL (7) singlets.) The Weyl transformation brings this to since the external coordinate x 6 is invariant under the Weyl transformation w 11 . This is part of a field strength of type F 10,3 = * 11 F 1 3 [84] that decomposes into a singlet under SL(7) and the 10 of SL(4) that is conjugate to the flux of the 5 3 in the 10.
We can also rephrase this in its dual version as follows. The original singlet flux becomes, using the background metric, dual to the components F abcd since and thus can be thought of as deriving from the three-form potential A abc depending on the coordinates y a =Ŷ a5 , from which we obtain the singlet flux F abcd = 4∂ [a A bcd] . At this point we can consider the M5 brane to be unsmeared. If we perform the U-duality transformation U 123 on this configuration, the potential A abc is mapped to the dual non-geometric potential Ω abc while the geometric coordinateŝ Y a5 are mapped to the non-geometric coordinatesŶ a4 (except forŶ 45 = y 4 = x 7 that stays invariant). The singlet flux is mapped to a flux living in the 10 as needed for the 5 3 brane.
To understand our use of dual potentials and electric couplings, it might be illuminating to consider the dual DFT discussion in [43,85]. This is possible since the whole construction takes place in the NSNS sector. The usual six-form (to which the NS5 couples) comes from the g −2 s potential D M N OP when decomposed under SL (10). We see from the above equation that we can describe the same flux either using the two-form or the four-form but they require dual derivatives for doing so. This is also consistent with a T-duality transformation on a DFT object acting on both the potentials and the coordinates at the same time. This means that, for example, the H-flux Hαβγ transform under duality on theα = 1, · · · , D index as Both of these contribute to the F -flux. One obtains similar possibilities for the action when continuing on to the Q-and R-fluxes.

Conclusions
In this work, transformations of the SL(5) U-duality group have been considered within the framework of EFT-a theory living on an extended spacetime parametrized by 7+(4+6) coordinates. Of these, 7 are the so-called external coordinates, 4 are geometric coordinates and 6 are the coordinates corresponding to the winding modes of M2-brane. A set of solutions of this EFT source generalised fluxes just as the membrane and five brane source fluxes in eleven dimensional supergravity. These generalised fluxes transform covariantly under the SL(5) of the EFT.
After obtaining the SL(5) orbits for the generalised fluxes we show how certain flux components transform into each other. As an explicit example of this we considered the background of the KK6-monopole and perform an SL(5) rotation to obtain the 5 3 -brane. This solution is clearly non-geometric as it depends on 3 non-geometric (dual) coordinates and generates a non-geometric flux, which has been explicitly calculated.
These SL(5) orbits are in perfect correspondence with the results obtained in [63] which were obtained by lifting the Buscher rules of string theory. In this work, we present a systematized approach based on the root systems of the corresponding U-duality algebra and the weight systems of the irreducible representations contributing to the embedding tensor. This has two advantages; firstly, it allows us to consider more general transformations than those considered in [63] and secondly, it allows us to consider other U-duality groups. In an upcoming paper [86] we investigate a similar story for the EFT with group E 7 (7) .
The results obtained here are the EFT analogue of the DFT description of T-duality orbit that generate so called Q and R fluxes as first described in Blair:2014zba.
Hαβγ −→ fαβγ −→ Qαβγ −→ Rαβγ (6.1) Although this was originally investigated with a toy model starting with H-flux on a torus (obviously not a true string background) they have since been realized by a series of genuine solutions of the DFT equations of motion, allowing for a dependence on non-geometric coordinates [51]. Since such backgrounds can no longer be solutions of the equations of motion of conventional supergravity, the dependence on winding coordinates is an essential stringy feature. Indeed, such a dependence is known to be related to worldsheet instanton corrections, which are non-perturbative corrections to the twodimensional σ-model [44,47]. Hence, one may speculate that the backgrounds of EFT, with fields depending on non-geometric coordinates, should also have some relation to non-perturbative corrections coming from the worldsheet of M2-(and/or M5-)branes. This is also discussed in [54] and we hope to investigate this in the future. Apart from the obvious extension to the E 7(7) U-duality group, we aim to study the Bianchi identities which are known to be crucial for string cosmological model-building. The work [69] presents the Bianchi identities for the so-called P -fluxes, i.e. those encoded in the vector-spinor θ M A of DFT, which are constant. See [62] for recent work on the socalled P -fluxes in EFT. Understanding the origin of the fluxes of the Type II theories from the full U-duality covariant generalised flux, one is able to recover coordinate-dependent fluxes which include,not only fluxes of g −3 s branes, but also branes with lower powers of g s . This is work in progress.
The generalised flux of DFT allows one to define the magnetic charge of the corresponding brane by a straightforward generalization of the usual Yang-Mills magnetic monopole charge [87] µ ∝ F M N K dX M ∧ dX N ∧ dX K , (6.2) which, indeed, gives the expected results. The generalization to the embedding tensor of EFT is not so simple. The main problem is that the irreducible representations of the corresponding embedding tensor do not contain scalars upon contraction with dŶ M ∧ dŶ N ∧ dŶ K . In contrast, one has to increase the number of wedged dŶ 's in the integrand and project them onto a specific representation. Although straightforward, this is not a technically simple calculation and the problem of magnetic charges for exceptional nongeometric branes needs to be investigated separately.
As has been discussed in detail in the previous section, there is an open problem of relating the results which one obtains by rotating EFT backgrounds with the results coming from the E 11 decomposition and more general U-duality actions on gauge potentials. So far, this is an open question which will hopefully be clarified in [86]. Note, in this paper, we have used the terminology 'non-geometric' to refer to configurations that cannot be written in terms of the usual supergravity fields in a way that they depend only the usual geometric coordinates. All the calculations we have performed were local in extended space. It is an interesting open question to show explicitly that the global structure of the solutions is such that they requiring transition functions that are not standard diffeomorphisms but more general elements of the EFT group, similar to what happens in [88,89,90,91,92]. These are fascinating global questions. In [93,94,95,96] these global questions were addressed and various proposals made for DFT. A clear input (especially in [94,95]) was from the study of so called topological T-duality of the type studied in [97]. The work in this paper, although only local, gives hints at how topological T-duaity might be generalised to a topological U-duality since the EFT solutions we describe contain fluxes which one imagines would be associated with some topological data. This is far from the scope of this paper but we hope it may inform an M-theory extension of [97].
Note added: While this manuscript was being finalised, the preprint [98] appeared that addresses related questions, in particular the orbit structure for SL(5) EFT.