Weaving the Exotic Web

String and M-theory contain a family of branes forming U-duality multiplets. In particular, standard branes with codimension higher than or equal to two, can be explicitly found as supergravity solutions. However, whether domain-wall branes and space-filling branes can be found as supergravity solutions is still unclear. In this paper, we firstly provide a full list of exotic branes in type II string theory or M-theory compactified to three or higher dimensions. We show how to systematically obtain backgrounds of exotic domain-wall branes and space-filling branes as solutions of the double field theory or the exceptional field theory. Such solutions explicitly depend on the winding coordinates and cannot be given as solutions of the conventional supergravity theories. However, as the domain-wall solutions depend linearly on the winding coordinates, we describe them as solutions of deformed supergravities such as the Romans massive IIA supergravity or lower-dimensional gauged supergravities. We establish explicit relations among the domain-wall branes, the mixed-symmetry potentials, the locally non-geometric fluxes, and deformed supergravities.


Contents
1 Introduction and summary 1

.1 Background
The ten-dimensional type II superstring theories contain a rich variety of extended objects such as the D-branes and the NS5-brane. The tension of a Dp-brane is proportional to g −1 s (g s : string coupling constant) and that of the NS5-brane is proportional to g −2 s . It is conjectured that string theories are related by discrete non-perturbative dualities.
When we compactify M/string-theory to lower dimensions, the U-duality group is enlarged and can relate objects that were not related in higher dimensions. That is to say, it can occur that, by a duality transformation, an extended object is mapped to a "non-geometric" one, being the latter an object that is not a solution of the higher-dimensional supergravity theories. This is a consequence of the geometric formulation of supergravity theories: the transition functions that are needed to "glue" the patches of the manifold on which the theory is defined do not contain the U-duality group transformations. Moreover, if we compactify the theory, the tension of the dualized extended object can change and be proportional to g α s with α ≤ −3. These objects are known as the exotic branes [1][2][3][4][5][6][7], and in this paper we are going to revisit various aspects of them.
Conceptually, exotic branes should not be considered that exotic: because they are obtained by duality transformations, their role is as important as the standard branes. Usually, the charges of these solutions are determined by the non-trivial monodromy that appears when we go around them. For example, let us consider type II supergravity on a T 2 with an NS5 brane extended along the external directions. The resulting background obtained after performing two T -dualizations is the so-called 5 2 2 -brane background. This background exhibits a non-trivial monodromy when going around the brane, which is not captured by the symmetries of the supergravity theories. The flux induced by this background, the so-called Q-flux, is the field strength of an antisymmetric object β mn , which is related by T -duality to the Kalb-Ramond B-field. That is to say, the 5 2 2 background is the source of the Q-flux, as it is magnetically coupled to the potential β mn .
Just like the Dp-brane is electrically coupled to the Ramond-Ramond (R-R) (p + 1)-form potential, in general, exotic branes are electrically coupled to mixed-symmetry potentials. In a series of works [8][9][10][11][12][13][14][15][16][17], it has been shown that there exists a one-to-one correspondence between exotic branes and some mixed-symmetry potentials that are defined in ten dimensions. A classification of mixed-symmetry potentials (and thus of exotic branes) has been done by considering different arguments. Firstly, the E 11 conjecture [18][19][20][21] allows one to predict directions are determined by a set of Killing vectors, which become crucial in the description of these massive or deformed supergravities [66,67]. The first formulations of these theories were prior to the DFT/EFT formulations. The case of the GSE [61,62] and its derivation from DFT [63,64] or EFT [65] is one of the most recent examples that have been worked out in the literature.
As mentioned above, because of their isometry directions, these deformed supergravities can be understood as effective lower-dimensional theories with massive deformations. Such deformations can be studied systematically: using the embedding tensor formalism [68,69] and constructing the tensor hierarchy of a theory, one can scan all the possible deformations of a particular lower-dimensional supergravity. Then, a dictionary between the fluxes associated to these deformed supergravities and the embedding tensor is estimated.
It is the purpose of this paper to establish a systematic way of studying the exotic branes and their expected-to-be one-to-one related objects. Based on the above arguments, we guess that the distinct formulations of exotic branes, mixed-symmetry potentials and the massive supergravities are closely related. We would like to fill the gaps among these three approaches and establish precise mechanisms to show their equivalence.
In this paper, we firstly generate the full web of exotic branes by applying U-duality transformations to standard branes. We only consider a subgroup of the U-duality group, which consists of the T -and S-duality transformations. As we could expect from the finiteness of the U-duality group for d ≥ 3, we have obtained a finite set of exotic states, which have been classified into different orbits.
After fully determining the web of branes, we find a systematic way to generate the exotic-brane backgrounds as solutions of duality-symmetric theories, namely DFT or EFT.
Being a U-duality-symmetric theory, EFT is the appropriate framework where to describe these backgrounds. Starting from a fully geometric brane background, we should be able to perform T -and S-duality transformations to generate the dualized backgrounds. To do so, we need to understand how duality transformations act on both the fields that enter the EFT and on the winding coordinates. That is to say, firstly, we rewrite the usual T -and S-duality rules in terms of the supergravity fields that appear in the M-theory and type IIB parameterizations of EFT. Secondly, we apply the duality transformations on the generalized coordinates. Because we start from geometric solutions that correspond to the standard branes, the sections (i.e. solutions of the SC) for the obtained solution are T -and S-dualityrelated to the geometric section.
The dictionary between the supergravity fields and the dual (or the non-geometric) fields in DFT/EFT allows us to calculate the non-geometric fluxes. We find a relation between the non-geometric fluxes and the mixed-symmetry potentials obtained from the E 11 decomposition and the wrapping rules.
Finally, we find a mechanism to systematically obtain ten/eleven-dimensional deformed supergravities that exhibit some isometry directions. The number of isometry directions depends on the specific solution of the SC, which will pick the non-physical coordinates that the fields can depend on. In general, we would like to engineer a systematic way to generate deformed supergravities that contain each of the exotic domain-wall branes as a solution. For instance, a relation between the domain-wall solutions and deformed supergravities has been suggested in [14].

Main results
In this subsection we summarize the results that we have obtained.
In Section 3, by brute force application of the S-and T -duality transformations, we have generated the full web of supersymmetric branes for each p-brane multiplet at any dimension d ≥ 3. In this classification, we have distinguished the defect, domain-wall and space-filling brane types (which have codimension 2, 1, and 0, respectively) from the standard branes.
In (3.13), (3.14), and (3.15), we have shown the spectrum of all the M-theory branes, type IIA branes, and type IIB branes. In Figures 3.1-3.20, we have generated the web of type II branes and shown the T -duality and S-duality chains of transformations that relate them. At any dimension d ≥ 3, we have obtained the spectra of exotic branes for any p-brane multiplet together with their degeneracies, which are given in Appendix C.
In Section 4, by utilizing the manifest O(d, d) T -duality symmetry of DFT, we have obtained some known domain-wall solutions, the D8 solution and the 5 3 2 solution (also known as the R-brane solution). In order to obtain the backgrounds of the full web of branes, T -duality is not enough. That is to say, to generate the whole T -duality orbits of Figures 3.1-3.20 which only contain domain-wall and space-filling branes, we additionally need S-duality transformations. By S-dualizing some elements of the orbits containing standard and defect branes, we generate the orbits spanned by domain-wall and space-filling branes.
Then, in Section 5, by making use of S-and T -duality transformations, we have obtained the full web of branes as solutions of EFT. We have shown that to obtain the exotic branes as EFT solutions, we have to systematically apply S-and T -duality transformations on both the EFT generalized metric and the set of coordinates defined in EFT. Unlike the well-known defect-brane solutions, the obtained solutions include the explicit linear winding-coordinate dependence, while satisfying the SC.
In terms of the dual fields in EFT, we have calculated all the non-geometric fluxes for each exotic brane in the web. Similar to the fact that the defect branes can be regarded as the magnetic sources of the globally non-geometric fluxes, the domain-wall branes are identified as the magnetic sources of the locally non-geometric R-fluxes. We have proposed suitable definitions of the R-fluxes in the E 8 (8) EFT that transform covariantly under Uduality transformations. Then, we have shown that the domain-wall-brane solutions in EFT contain constant R-fluxes.
We have clarified the relation between the non-geometric fluxes associated to each brane and the mixed-symmetry potentials predicted in the literature. In particular, we have shown that the non-geometric fluxes in EFT are dual to the field strengths of the mixed-symmetry potentials. To do so, we have to extend the electric-magnetic duality transformation of the mixed-symmetry potentials that was conjectured in DFT [70] to the EFT formulation, for both the M-theory/IIB parameterizations. The electric-magnetic duality transformation in EFT involves the dual spacetime metric, as it occurs in the DFT case.
Finally, in Section 6, we have discussed various deformed supergravity theories, which generalize the Romans massive IIA supergravity. As mentioned earlier, these theories can enjoy one or more isometry directions, each of them characterized by a Killing vector. For a given exotic brane, we have provided a prescription to identify the lower-dimensional deformed supergravity theory that realizes that background. While standard and defect branes do not exhibit any dependence on winding coordinates, this is not the case for the domain-walls and the space-filling branes. The winding-coordinate dependence of the domain-wall solutions are transmuted into the R-fluxes (or the gaugings), which characterize the deformations of the supergravities, and the domain-wall solutions in the deformed supergravities are independent of the winding coordinates. That is to say, the dependence on the winding coordinate is encoded in the deformation parameter of the corresponding supergravities. We have reproduced several known domain-wall solutions in certain deformed supergravities, which include known solutions in [67].
In summary, in this paper, we have explicitly established one-to-one mappings among several topics, mixed-symmetry potentials ↔ direct manner. In EFT, there exists an external 1-form A I 1 µ that transforms in the particle multiplet, and a 2-form B I 2 µµ that transforms in the string multiplet, and a 3-form C I 3 µµρ that transforms in the membrane multiplet, and so on. As we have explicitly shown in this paper, these p-brane multiplets include all of the exotic branes and these (p+1)-form fields should be composed of the mixed-symmetry potentials. The explicit parameterizations of these (p + 1)form fields will be important. More importantly, there must be constraints for the derivative of the (p + 1)-form fields and the derivative of the generalized metric, corresponding to the electric-magnetic duality between the field strength of the mixed-symmetry potentials and the R-fluxes. Such duality will correspond to the exotic duality [78] recently discussed in DFT, and it is important to identify the duality relation in EFT.
As a different direction, it would be interesting to determine the worldvolume actions for exotic branes and show that the Wess-Zumino term indeed contains the mixed-symmetry potential (see [8] for a related work).
Let us comment on the R-fluxes. Despite in this work we have given some heuristic definitions of the locally non-geometric R-fluxes, we should provide more systematic definitions similar to [79][80][81]. Here, we have concentrated on the R-fluxes associated with the "elementary" domain-walls, but since the mixed-symmetry potentials for the missing states are also proposed, by performing the electric-magnetic duality, we may also obtain the R-fluxes associated with the missing states. Only after introducing such R-fluxes, we can obtain a U-duality multiplet of fluxes.
It is also interesting to study the fluxes associated with the space-filling branes. In terms of the mixed-symmetry potentials in type II theory, mixed-symmetry potentials with a set of ten antisymmetric indices, such as E 10,7−p,2 have been proposed. Naively, by introducing their field strengths and performing the electric-magnetic duality, one may obtain the corresponding fluxes. However, proper definitions of the field strengths are not clear at present. Some hints may be found by studying the EFT solutions of the space-filling branes in more detail.
In this paper, we have made clear how to obtain the action or the equations of motion of various deformed supergravities from EFT. According to the SC in EFT, the deformed supergravities are effectively defined in lower-dimensional spacetime, as some of the winding coordinates are used to provide the constant fluxes or deformation parameters. It would be interesting to establish a systematic relation between the exotic branes or, equivalently, their associated lower-dimensional deformed supergravities and the gaugings in the language of the embedding tensor. Some work in this direction has been recently done for spacetime-filling branes in [82]. In this paper, among all of the domain-wall branes contained in the U-duality multiplets, we have only considered the "elementary" domain-walls. In that case, the SC is not violated. On the other hand, in [83], gaugings that break the SC have been found.
In particular, non-geometric fluxes that are in different orbits from the standard fluxes are introduced. It would be relevant to find a connection between these fluxes and the missing states in the U-duality multiplets.

Plan of the paper
The paper is organized as follows. In Section 2 we briefly review the notation for exotic branes.
In Section 3, we review the relation between exotic branes and the weights of the U-duality group. We then construct the full web of "elementary" exotic branes and give the duality transformations that relate them. In Section 4, we review how to obtain some of the known domain-wall backgrounds as solutions of DFT. In Section 5, we explain how to obtain all the "elementary" exotic branes as solutions of EFT. The definitions of the non-geometric R-fluxes are also provided. In Section 6, we review some deformed/massive supergravity theories and show that they can be obtained upon solving the SC of DFT/EFT in such a way that winding coordinates are allowed.
We also provide several appendices. Appendix A provides the notation used along this work. In Appendix B, we review parameterizations of the generalized metric in E n(n) EFT (n ≤ 7). Appendix C shows the various spectra of p-brane multiplets for diverse dimensions.
Finally, Appendix D shows the relation between the exotic branes that we have obtained and the mixed-symmetries potentials that are coupled to.
Addendum. Upon publication of version 1 of this work, we learned that some aspects of Section 3 were also constructed by another group [131].

A brief review of duality rules and exotic branes
In this section, we provide a brief review of exotic branes in type II string theories and M-theory toroidally compactified to d-dimensions.

Type II branes
In type II string theory, by denoting the radius of the torus along the x i -direction as R i , the mass of a fundamental string (denoted as F1) wrapped along the x i -direction is given by By using the familiar T -and S-duality transformation rules, S-duality : g s → 1/g s , l s → g 1/2 s l s , we can see how the mass (2.1) is transformed under duality transformations. For example, if we perform a T -duality along the x i -direction, the mass (2.1) becomes which is interpreted as a mass of the pp-wave or the Kaluza-Klein (KK) momentum (denoted as P). If we instead perform an S-duality, the mass (2.1) becomes which is interpreted as a mass of the D1-brane wrapped along the x i -direction. By repeating duality transformations, we obtain masses of various branes. It is then useful to employ the notation of [5] (see also [84,85]), which allows us to characterize various branes by their masses. If an object wrapped along x n 1 , . . . , x n b -directions has a mass, we denote the brane as b (cs,...,c 2 ) n (n 1 · · · n b , m 1 · · · m c 2 , · · · , p 1 · · · p cs ) , (2.6) or simply call it a b (cs,...,c 2 ) n -brane. 1 With this notation, for example, the usual Dp-brane and the NS5-brane are denoted as the p 1 -brane and the 5 2 -brane.

M-theory branes
We can uplift the mass of type IIA branes to the mass of M-theory branes by using the usual relation connecting 11D and 10D, where R M represents the radius of the M-theory circle. After the uplift, M-theory branes generally have masses of the form (2.8) 1 We denote b We then denote the brane as b (cs,...,c 2 ) n (i 1 · · · i b , j 1 · · · j c 2 , · · · , k 1 · · · k cs ) , (2.9) where n ≡ 1 + b + 2 c 2 + 3 c 3 + · · · + s c s represents the power of the Planck length in the denominator. In the literature, n is omitted, but here we keep it since it is a good measure of the exoticism, similar to the power of g s in type II theory.
In terms of M-theory, the transformation rule (2.2) can be nicely summarized as [1,86] U i,j,k : It is noted that the inverse of the Newton constant in d-dimensions, is invariant under the U i,j,k for arbitrary choices of three directions {i, j, k} in the torus T n .
As we review in Section 3.1, T -and S-duality in type II theory, or the duality transformation U i,j,k in M-theory can be regarded as the Weyl reflection associated with the simple roots of the E n(n) group, which is the U-duality group of the string/M-theory. In the next section, we provide a full list of branes obtained by the U-duality transformations.
3 Full duality web for d ≥ 3 In this section, we provide the full duality web for string/M-theory compactified to d-dimensions with d ≥ 3.

Duality rotations as Weyl reflections
Before showing the duality web, here we explain that the chain of T -and S-duality can be regarded as Weyl reflections by closely following the discussion of [1,5].

Setup
Let us rewrite the brane tension (2.13) as which has the mass dimension and reduces to the mass (2.8) when p = 0. We also define a vector v ≡ v µ e µ (µ = 0, 1, . . . , n) by using a basis e µ of (n + 1)-dimensional vector space endowed with an inner product, Then, a particular U-duality transformation U 1,2,3 of (2.10) can be realized as a reflection,

4)
A general U-duality transformation (2.10) can be realized by combining U 1,2,3 and particular U-duality transformations P i : R i ↔ R i+1 (i = 1, . . . , n − 1), 2 and P i can be also realized as a In type II theory, by denoting a chain of T -dualities along the x m1 , . . . , x mn -direction by T m1···mn , a chain of dualities T m S T mn S T nm S T m , corresponds to a permutation R m ↔ R n keeping g s and l s invariant.
Therefore, an exchange R m ↔ R n in 11D can be also realized as a combination of U i,j,k . Furthermore, the 11D uplift of T m S T m corresponds to a permutation R m ↔ R M . Therefore, the U -duality (2.10) contains all possible permutations R i ↔ R j in 11D.

(3.8)
Since the vector δ is orthogonal to all of the simple roots, there is an ambiguity in the choice of λ i ; λ i ∼ λ i +c i δ (c i : constant). We can determine the constants c i by requiring α i = a ij λ j , but the δ-direction is irrelevant for our purpose, and we can mod out the direction from the (n + 1)-dimensional space spanned by e µ . With the above choice, we have

p-brane multiplet
According to the relation (3.1) between the tension and the vector, the tension associated with a fundamental weight λ 1 is T 1 = R 1 /l 3 p . This is the tension of a string in the external d-dimensional spacetime. More concretely, this string can be interpreted as an M2-brane wrapped along the internal x 1 -direction. By acting E n(n) U-duality transformations, we obtain the U-duality multiplet, known as the string multiplet, that is associated with the fundamental weight λ 1 .
The tension associated with λ 2 is T 3 = R 1 R 2 /l 6 p , which corresponds to the tension of a 3-brane in the external d-dimensional spacetime. In terms of M-theory states, it is an M5brane wrapped along the internal x 1 and x 2 -directions. Performing E n(n) transformations, we obtain the 3-brane multiplet. Note that in order to consider the 3-brane multiplet, the dimension d needs to satisfy d ≥ 4 .
Similarly, the tension associated with λ 3 is T ′ 5 = R 1 R 2 R 3 /l 9 p and it makes a certain 5brane multiplet. If there is an "M8-brane" that has a tension 1 lp(2πlp) 8 , the tension can be interpreted as the tension of an M8-brane wrapped along the internal x 1 , x 2 , and x 3 -directions.
However, the existence of such object is not clearly understood. The tensions associated with λ i (i = 4, . . . , n − 2) do not have a clear interpretation either.
The tension associated with λ n−1 = δ − e n ∼ −e n is T 0 = 1/R n . This can be regarded as the mass of the pp-wave, and the corresponding multiplet is called the particle multiplet.
Finally, the tension associated with λ n is T 2 = l −3 p which is nothing but the tension of the M2-brane. The corresponding multiplet is known as the membrane multiplet. This completes the fundamental representations of the E n(n) U-duality group.
We can also consider the 4-brane multiplet by considering a tension of the M5-brane wrapped along the x 1 -direction, T 4 = R 1 /l 6 p . This corresponds to a weight λ (4) = e 1 − 2 e 0 = λ 1 + λ n . Thus, the 4-brane multiplet is the representation labelled by the Dynkin label The tension is given by T (KKM) 5 = R 2 1 R 2 /l 9 p and it corresponds to λ (KKM) = λ 1 + λ 2 . Namely, the second 5-brane multiplet has the Dynkin label [1, 1, 0, . . . , 0]. Higher p-brane multiplets can also be constructed similarly. We can summarize this subsection with As an example, let us consider a string multiplet in M-theory compactified on T 6 , where the Uduality group is E 6(6) . We start from the highest weight vector [1, 0, 0, 0, 0, 0] that corresponds to a 2 3 -brane wrapped along the x 1 -direction. In order to indicate that the 2 3 -brane behaves as a string in the external five-dimensional spacetime, we denote it as 2 3 (·1)-brane, where the dot " · " corresponds to one external dimension. As described in Table 3.2, by subtracting the simple roots, we can obtain the weight diagram for the 27-dimensional string multiplet.
Here, let us briefly explain how to make the identification between the Dynkin labels and  Table 3.1: Dynkin labels of the p-brane multiplets, and the tension associated with the highest weight vector and the associated M-theory brane.
the brane charges. When we subtract a simple root α n = e 0 − (e 1 + e 2 + e 3 ) , the brane tension is multiplied by R 1 R 2 R 3 /l 3 p . At the same time, the Dynkin label is reduced by [a n1 , . . . , a nn ] corresponding to α n = a nj λ j . Similarly, when we subtract α k = e k −e k+1 = a ki λ i (k = n−1), the brane tension is multiplied by R k /R k+1 and the Dynkin label is reduced by [a k1 , . . . , a kn ] .
On the other hand, when we subtract α n−1 = e n−1 − e n = a (n−1)j λ j − δ, the brane tension is multiplied by R n−1 /R n . In this case, the Dynkin label is reduced by [a (n−1)1 , . . . , a (n−1)n ] and the information about δ, which corresponds to the inverse of the d-dimensional Newton constant l d , is lost. Accordingly, when we try to reproduce the tension from the Dynkin label, we should introduce l d appropriately. For example, the Dynkin label [0, −1, 1, 0, 0, −1] (3.10) In order to make the mass dimension the same as that of the string tension, we multiply it by l 5 and obtain By using the convention (2.13), this is interpreted as the tension of the 6 1 9 (·12456, 3)-brane. From a similar consideration, we can find the identifications between the Dynkin labels and branes shown in Table 3.2.

Web of supersymmetric branes
Utilizing the duality transformation rule (2.10), we can generate a chain of exotic branes in M-theory. Indeed, by brute force applications of duality (2.10) to the tensions of the standard branes, we obtain Tables C.1-C.9, which show the explicit brane charges and the degeneracies in each multiplet. By summing up the degeneracies of all branes, we obtain the size of the Weyl orbit in each p-brane multiplet in d-dimensions, as summarized in Table 3  of (b + c 2 + · · · + c s )-brane, and the codimension is given by 10 − (b + c 2 + · · · + c s ). If the codimension of a brane is equal to 2, 1, or 0, we call it a defect brane, a domain-wall brane, or a space-filling brane, respectively. These are also called the non-standard branes while branes with codimension 3 or greater are called the standard branes. For clarification, in (3.13), we have colored the defect branes, the domain-wall branes, and the space-filling branes in purple, blue, and darkcyan, respectively.
As we can see from Table 3 In the previous subsection, we have shown the Dynkin labels only for p-brane multiplets with p ≤ 5 . In d ≥ 7 , we also have the p-brane multiplet with p = 6, 7, 8 . For the 6-brane multiplet, the highest weight corresponds to a 6 1 9 -brane with the Taub-NUT direction given by the x 1 -direction. The tension is T 6 = R 2 1 /l 9 p and it corresponds to 2 λ 1 + λ n , whose Dynkin label is [2, 0, 0, 1] in d = 7 and [2, 0, 1] in d = 8 . For the 7-brane multiplet, the highest weight corresponds to a 8 (1,0) 12 -brane with the special isometry direction given by the x 1 -direction and wrapped along the x 2 -direction. The tension is T 7 = R 3 1 R 2 /l 12 p , corresponding to 2 λ 1 + λ 2 and [2, 1, 0] in d = 8.
are known as the higher KK branes denoted as Dp 7−p [7]. In addition, 7 (1,0) 3 is known as the KK8A-brane in [67]. As one can clearly see, in dimensions d ≥ 3 , there exist the type IIA branes with tensions proportional to g α s with −11 ≤ α ≤ 0 . In order to obtain all of the "elementary" type IIB branes, we act a T -duality to each of the type IIA branes. Since a T -duality does not change the power of g s , the type IIB branes also have tensions proportional to g α s with −11 ≤ α ≤ 0 . A list of all of the "elementary" type IIB branes is as follows: We can also summarize the T -duality web between the type IIA branes (upper) and the type IIB branes (lower) as in Figures 3.1-3.20:       Figure 3.7: T -duality chain of the E (4;0) -branes.             [ Figure 3.20: T -duality chain of the E (11;7) -branes.
Here, for the S-duality non-singlets in the type IIB side, we have appended the subscript with round brackets. For example, 2 (7,0,0,0) 11 (8) in Figure 3.20 represents the 2 (7,0,0,0) 11 -brane, and also denotes that its S-dual partner is the 2 (7,0,0,0) 8 -brane. The characters in the squared brackets are not important here, and will be explained in Section 5.9. Each (solid or dashed) line corresponds to a T -duality and the circled numbers have the following meaning. For example, the 5 3 2 -brane in Figure 3.3 has three types of direction along which we can perform T -duality; (i) directions along which the mass does not depend on the radii, which we call 0 , (ii) directions, denoted as 1 , along which the mass linearly depends on the radii, (iii) three directions denoted as 2 along which the mass quadratically depends on the radii. If we perform a T -duality along the 0 direction, we obtain the 5 4 2 -brane, while if we perform a T -duality along the 1 direction, we obtain the 5 3 2 -brane, and along the 2 direction, we obtain the 5 2 2 -brane. Namely, the circled number can be understood as the power of the radius dependence along which the T -duality is performed. The meaning between the solid or dashed line, which is not important here, is that each line connected to even/odd number in the type IIB side is a solid/dashed line.

Web of the missing states
In the previous subsection, we have only considered the branes that are connected to the standard branes via U-duality transformations (2.10). However, as we can clearly see from Table 3.3, if we consider the non-standard branes (i.e. colored branes with codimension 2 or less), these are not enough to make up the U-duality multiplet. We need to introduce additional states, which we call missing states for obvious reason.
The existence of the missing states was originally noted in [1], and they were later discussed for example in [22,73,74]. Properties of such missing states are not clearly understood, and they may not be supersymmetric states as conjectured in [22]. Here, we only compute the tensions of these states by simply extrapolating the correspondence between tensions and Dynkin labels discussed in the previous sections to arbitrary weight vectors (see [87] for a similar work in the context of E 11 ).

Example: 4-brane multiplet in E 4(4)
Let us start with a simple example, a 4-brane multiplet in M-theory compactified on T 4 . In this case, Table 3.3 shows that the number of supersymmetric branes is 20, although the dimension of the 4-brane multiplet is 24. Thus, there are four missing states. In order to identify the missing states, let us consider the weight diagram for the 4-brane multiplet given in Table 3.5.
Since the mass dimension of the tension T 4 is five, the four degenerate Dynkin labels [0, 0, 0, 0] correspond to (3.16) By using the convention (2.13), this should be understood as a tension of the 8 9 (· · · · 1234)brane, where the four dots · · ·· represent that the 8-brane is extended along certain four external spatial directions. This kind of 8-brane was predicted in [1] and called M8-brane in [3], although its properties are unclear so far. We can just extrapolate their tensions. From the tension, we can find that these states are singlets under the Weyl reflections.
column have the weights with the same length, and we have checked that they are indeed in a single U-duality orbit of (2.10).
In terms of M-theory, the following states are contained in Tables C.10-C.16: One can also make a list of type II branes appearing in Tables C.10-C. 16 and draw a duality web along these states.
As we can see from Tables C.10-C. 16, the missing states in the p-brane multiplet have degeneracies which depend on p . For example, the 8 9 -brane in the p-brane multiplet (1 ≤ p ≤ 6) has degeneracy (8 − p) , although for p = 6 the degeneracy becomes 1. The p-dependence is non-trivial, but the degeneracy is independent of d for all missing states. The missing states in higher d can be obtained from the missing states in lower d just by truncating the states that are disallowed by dimensionality.
For n = 1 (KK monopole), the potential D 7,1 is the magnetic dual of the graviphoton, known as the dual graviton. The worldvolume action of the KK monopole including the Wess-Zumino term has been obtained in [89]. For n = 2, the potential D 8,2 is rather non-standard but it is known to be the magnetic dual of the so-called β-field. Its coupling to the 5 2 2 -brane has been determined in [90,91] (see also [92,125]). The M-theory uplift, the action for the 5 3 12 -brane was also studied in [93]. Generalizations to n = 3 and 4 in the manifestly T -duality covariant approach have been achieved in [94].
The S-dual of the 5 2 2 -brane is 5 2 3 -brane is a member of the E(xotic)-branes. The E-brane (p + n) (n,7−p−n) 3 couples to the mixed-symmetry potential E 8+n,7−p,n . In general, there is a conjectural relation between supersymmetric branes and the mixed-symmetry potentials: In the convention of [13], depending on the power of the string coupling n, the potentials are denoted as E (n = 3), F (n = 4), G (n = 5), H (n = 6), . . . . In this paper, since the integer n runs up to 11, we call them E (n) (i.e. E (4) = F , E (5) = G, E (6) = H, and so on) and denote the corresponding brane the E (n) -brane. In fact, as we can see from Table 3.6, for example, there are three families of E (4) -branes, and we distinguish them by introducing additional integers as E (4;6) , E (4; 3) , and E (4;0) . The general rule for the second integer is very simple; is a member of the E (2n;cn) -brane and an exotic brane b is a member of the E (2n+1;cn+c n+1 ) -brane.
The first set of indices in the mixed-symmetry potential E (n) m 1 ···m 1+b+c 2 +···+cs,··· corresponds 10,6+n,6+n,q,n,n    Table 3.6: Exotic branes with the tension proportional to g α s (α ≤ −3) and the corresponding mixed-symmetry potentials in type II string theories. Here, n and m are non-negative integers while p (q) runs over non-negative even/odd (odd/even) numbers in type IIA/IIB theory.
to the worldvolume directions of the brane, and the directions after the first comma correspond to the isometry directions, namely the internal toroidal directions. As it is suggested from the relation (3.17), in order to relate the supersymmetric branes to the mixed-symmetry potentials, a set of indices delimited by commas has to be a subset of the set of indices sitting to the left. For example, E 0µνρ1234,34 couples to the exotic 5 2 3 (µνρ12, 34)-brane while there is no supersymmetric brane which couples to E 0µνρ1234, 45 . By considering this argument, there is a one-to-one correspondence between supersymmetric branes and mixed-symmetry potentials. The explicit counting of the number of mixed-symmetry potentials in each dimension is summarized in Appendix D.

Exotic-brane solutions in DFT
In this section, we explain how to construct the supergravity solutions for the variety of exotic branes discussed in the previous section. If we consider only the standard branes or the defect branes, we can (at least locally) write down the solutions satisfying the standard supergravity equations of motion. However, as we discuss in this section, for domain-wall branes or spacefilling branes, we need to employ the manifestly duality-covariant formulations of supergravity, such as the DFT or EFT. This section is devoted to descriptions of exotic-brane solutions in DFT while the descriptions in EFT are discussed in Section 5.

D7-brane solution
Let us begin with the standard D7(1234567)-brane solution, where τ (x 8 , x 9 ) ≡ τ 1 + i τ 2 is given by  Here, we have introduced a shorthand notation, We now consider a domain-wall solution, the D8 solution. Since the D7 solution has codimension two, we need to implement the standard smearing procedure, which changes the function τ (x 8 , x 9 ) keeping the expression (4.1) intact. The resulting functions after the smearing are where h 0 and m are constants. In the case of the Dp-brane solution with p ≤ 6 that depends on the transverse (9−p) coordinates, the standard smearing procedure produces an additional isometry direction, and by performing a T -duality in the isometry direction, we can obtain the D(p + 1)-brane solution. However, in the case of p = 7, the smeared solution still depends on the two coordinates x 8 and x 9 and we cannot perform the usual T -duality to obtain the D8-brane solution. This is a new feature of the domain-wall solution.

A quick review of DFT
where we have defined a matrix e i ≡ diag(0, . . . , 0, The relation between the DFT fields and the usual supergravity fields is as follows. The generalized metric and the dilaton can be parameterized as 4 On the other hand, the O(10, 10) spinor |A is defined on the Clifford vacuum |0 satisfying Here, the gamma matrices (γ M ) ≡ (γ m , γ m ) are defined by and polyform). In type IIA/IIB theory, only the R-R odd/even-form potentials are included, and thus |A is defined to satisfy Here, γ 11 is defined as The field strength is defined as Unlike the standard supergravity fields, the DFT fields can depend on the generalized coordinates x M but the consistency condition, namely the SC, requires that the DFT fields cannot depend on more than ten coordinates out of twenty. If we keep the dependence on the standard coordinates x m , for example, the field strength (4.10) is reduced to the usual one, In this paper, we consider different choices of coordinates where supergravity fields depend on some of the winding coordinatesx m .

D8-brane solution
By using the above setup, let us construct the D8-brane solution in DFT. We start from the smeared D7 solution (4.1), and perform the formal T -duality (4.5) along the x 8 -direction. We then obtain Since the formal T -duality changes the coordinates x 8 ↔x 8 , the τ 1 here has the linear winding-coordinate dependence; τ 1 = mx 8 . In fact, this is precisely the D8 solution in DFT [59] (which corresponds to the familiar D8 solution [66]). The field strength becomes which means that the background has the constant 0-form and the dual 10-form field strengths The relation to the Romans massive IIA supergravity [58] is discussed in Section 6.
Let us start with the background of the (smeared) exotic 5 2 2 (12345, 67)-brane [67], where τ 1 = m x 8 and τ 2 = h 0 + m |x 9 | . Here, D 6 is the potential of the dual field strength H 7 ≡ dD 6 + · · · (where the ellipses denote irrelevant terms depending on type IIA or IIB) satisfying H 7 = e −2Φ * 10 H 3 . Again by performing a formal T -duality along the x 8 -direction, we obtain the background of the 5 3 2 (12345, 678)-brane, where τ 1 = mx 8 and τ 2 = h 0 +m |x 9 | . The DFT fields associated with these fields (g mn , B mn , Φ) satisfy the equations of motion of DFT as it is expected, as the formal T -duality always maps a solution to a solution.
In fact, in order to describe the 5 2 2 or the 5 3 2 backgrounds, it is more convenient to introduce the dual supergravity fields (g mn , β mn ,φ) suggested in [98,99]. They are defined through and can be regarded as redefinitions of the supergravity fields. More explicitly, we obtain 5 From the relation, we can determine the dual parameterization for the 5 2 2 background as [95] 20) and the 5 3 2 background as In the dual description, the winding-coordinate dependence in the 5 3 2 background is contained only in the β-field. Moreover, its dependence is only linear similar to the D8 background. In fact, as we show in the next section, in all of the "elementary" domain-wall solutions, the winding-coordinate dependence appears only in a certain gauge field linearly.

Non-geometric fluxes and mixed-symmetry potentials
In the dual parameterization, we can define the so-called non-geometric Q-flux [100][101][102][103] as The non-geometricity of the 5 2 2 -brane (or the Q-brane) background was pointed out in [84], and as shown in [95,104], the 5 2 2 (12345, 67) background has a constant Q-flux, Q867 = m . Here and hereafter, in order to avoid confusion, we may add bars on integers, like6 and7, indicating that these integers are associated with certain spacetime directions.
The low-energy effective Lagrangian for the non-geometric Q-flux was obtained in [99] as where |Q| 2 ≡g mng pq, rs Q m pq Q n rs , and we have usedg p 1 ···pn, q 1 ···qn ≡g p 1 r 1 · · ·g pnrn δ r 1 ···rn q 1 ···qn and δ r 1 ···rn qn] , andR is the Ricci scalar associated withg mn . The equation of motion for the β-field takes the form ∂ m e −2φ −gg mng pq, rs Q n rs = 0 , (4.25) and this suggests to introduce the dual field strength as [92] Q 9,2 ≡ e −2φg pq, rs * 10 Q pq Here, the subscript "9, 2" represents that the field strength is the mixed-symmetry tensor with 9 antisymmetric indices and 2 antisymmetric indices, and the Hodge star operator * 10 is associated with the dual metricg mn . By introducing the associated potential Q 9,2 ≡ dD 8,2 , we can find a connection between the non-geometric Q-flux and the mixed-symmetry potential D 8,2 introduced in a series of works [8][9][10][11][12][13][14][15][16][17] (see [12,70] for a similar Hodge duality between Q-flux and the mixed-symmetry potential D 8,2 ). In the 5 2 2 (12345, 67) background, we obtain As discussed in [95], by T -dualizing the Q-brane background, we can obtain the background of the R-brane, which is nothing but the 5 3 2 -brane. By defining the non-geometric R-flux, we can show that the 5 3 2 background contains a constant R-flux, The R-flux is sometimes called the locally non-geometric flux. In [105], the effective Lagrangian for the R-flux was derived from the DFT Lagrangian as (see also [71,72,[128][129][130]) where |R| 2 ≡ 1 3!g m 1 m 2 m 3 , n 1 n 2 n 3 R m 1 m 2 m 3 R n 1 n 2 n 3 . This again suggests to define the dual field strength as m 1 m 2 m 3 , n 1 n 2 n 3 * 10 R n 1 n 2 n 3 . (4.31) By defining the corresponding potential R 10,3 ≡ dD 9,3 , we obtain in the 5 3 2 (12345, 678) background. A similar duality relation between the mixed-symmetry potential D 9,3 and the R-flux was recently discussed in [70]. As it has been discussed there, background, our result is consistent with the above T -duality rule.
According to the above relation between Q-and R-fluxes and the mixed-symmetry potentials, we can summarize the famous T -duality chain [101] as follows:
Similar to DFT, it is defined on an extended spacetime with the generalized coordinates x I associated with the branes in the particle multiplet of the E n(n) group. In particular, when we consider M-theory/T n , the set of coordinates x I is parameterized as where the indices i, j, k run over the internal toroidal directions i = d, . . . , 11 and the external spacetime has the usual coordinates x µ (µ = 0, · · · , d−1). Each of the generalized coordinates corresponds to that of M-theory branes such as P, M2, M5, KKM etc., and the total number is equal to the dimension of the particle multiplet of the E n(n) group. On the other hand, when we consider type IIB theory/T n−1 , we can parameterize the same generalized coordinates as (x I (IIB) ) =(x m , y α m , y m 1 m 2 m 3 , y α m 1 ···m 5 , y m 1 ···m 6 , n , y αβ m 1 ···m 7 , y α m 1 ···m 7 , n 1 n 2 , y m 1 ···m 7 , n 1 ···n 4 , y α m 1 ···m 7 , n 1 ···n 6 , y m 1 ···m 7 , n 1 ···n 7 , p ) , supergravity (M-theory parameterization) and the bosonic fields in type IIB supergravity (type IIB parameterization). We refer to Appendix B for a more detailed study of these parameterizations. As determined in [106] for the cases E n(n) EFT (n ≤ 7), the two parameterizations can be related by a linear map, Here, the S I J is a constant matrix and under this transformation, the equations of motion of EFT (prior to choosing a particular solution of the SC) are transformed covariantly. If we rewrite the fields in 11D supergravity in terms of those in type IIA supergravity, by comparing both sides in (5.3), we find the standard T -duality rules between type IIA and type IIB supergravity [106]. Therefore, we can do T -duality transformations throughout the linear map (5.3), as the matrix S I J contains the information of the T -duality direction. On the other hand, the S-duality rule is rather trivial. In the type IIB parameterization, all of the generalized coordinates and the supergravity fields are SL(2) tensors, and the indices α and β are rotated by a matrix Λ α β ≡ 0 1 −1 0 as usual.

Dual parameterization in the whole bosonic sector
In the case of DFT, the conventional fields and the dual fields are related through the expression (4.18). Here, we briefly explain how to generalize the relation (4.18) to EFT.
As we already explained, the generalized metric M IJ in EFT can be parameterized by the bosonic fields in type IIB supergravity, which we call M (IIB) IJ . We can also parameterize the same generalized metric in terms of the dual fields in type IIB supergravity such as (g,φ, β mn , γ m 1 ···mp , · · · ) [106,107]. This is called the non-geometric parameterization since the dual fields are related to the non-geometric fluxes, and we call the generalized metric we can, in principle, determine the dual fields in terms of the conventional fields.
Since the generalized metric contains only the supergravity fields with internal (toroidal) components, for the metric with external indices g µν and g µm , we need a more elaborated recipe. For our purposes, it is enough to know the transformation rule for the components g µν . By truncating other external fields, the duality relation becomes [107] det g E where g E mn ≡ e − 1 2 Φ g mn andg E mn ≡ e − 1 2φg mn are internal components of the Einstein-frame metric that are contained in M IJ . We can compute the external components of the Einsteinframe dual metricg E µν and the string-frame metric is obtained asg µν ≡ e

Reorganization of the generalized coordinates
In order to simplify the T -duality rule, we here consider the following redefinitions of the generalized coordinates in type II theory. The winding coordinates for P and F1 (that appear also in DFT) are defined as   y m 1 ···m 6 , n n ∈ {m 1 · · · m 6 } , y 12 correspond to the eight missing states that are not connected to other branes under T -and S-dualities (see Section 3.3). In the following discussion, we do not consider these coordinates any more since these do not appear in our solutions.

First two examples of domain-wall solutions in EFT
Before considering all of the "elementary" domain-wall solutions, let us begin with two simple examples.

Dual parameterization for the 7 3 -brane solution
For the 7 3 background, the non-vanishing fields are the (g µν , g mn , Φ, C 0 ) . In this case, (5.4) and (5.5) are reduced to and we obtain the dual parameterization for the 7 3 -brane solution,

Dual parameterization for other p-brane solutions
We can similarly obtain the dual parameterizations for other p-brane solutions in (5.22) and (5.23). However, in general, a direct comparison of the generalized metrics is very complicated.

-brane background
As the second example, we consider the 1 (1,0,6) 4 -brane background. It can be obtained from the smeared 1 6 4 (1, 234567)-brane background: where τ is τ 1 = mx 8 and τ 2 = h 0 + m |x 9 | . Since there are no R-R fields, we can easily check that this is a solution of DFT.
The dual parameterization again can be obtained by comparing two parameterizations of the generalized metric in EFT, but in order to obtain the dual parameterization for the 1 6 4 solution, it is easier to S-dualize the 1 6 3 solution (5.26). The dual parameterization for the 1 (1,0,6) 4 solution can be obtained by further performing a T -duality. The results are as follows:

(5.45)
This kind of Hodge duality has been also suggested in [12]. The mixed-symmetry potential may be defined through Q 9,6 ≡ dE If we define the dual field strength as R 10,7,1,1 ≡ e −6φg m 1 ···m 7 , n 1 ···n 7g prgqs * 10 R m 1 ···m 7 , p, q ⊗ dx n 1 ∧ · · · ∧ dx n 7 ⊗ dx r ⊗ dx s , (5.48) and also define the potential through R 10,7,1,1 ≡ dE 9,7,1,1 } , whose field strengths are related to the magnetic fluxes as This suggests that there is a one-to-one correspondence between domain-wall branes, the 9form mixed-symmetry potentials, and the R-fluxes. Further, the set of indices {A} in the R-fluxes can be found from the set of indices {A} in the mixed-symmetry potentials, which are consistent with the general rule (3.17). In fact, this appears to be a general structure as we see below.
In the following, we will firstly introduce a generalization of the locally non-geometric R-fluxes, and then show that the domain-wall solutions in EFT have a constant R-flux.

Locally non-geometric fluxes
As we have already discussed, a domain-wall brane, say the b (cs,...,c 2 ) n -brane, is the magnetic source of the non-geometric flux with a set of antisymmetrized indices, R c 2 +···+cs,...,c s−1 +cs,cs , which is a U-duality version of the familiar R-flux (see [79][80][81]

Locally non-geometric fluxes in M-theory/T 8
We can easily uplift the R-fluxes obtained in type IIA theory to M-theory. The results are as follows: 15 -brane: -brane: -brane: -brane: -brane: -brane: (5.118) Note that the A m M is equal to the −γ m in type IIA theory while A M m is a complicated non-linear expression that will be related to R-R 1-form C m .
By using the identities such as (5.82), the fluxes R 4,1 , R 7,4 , and R 7,7 appear to be consistent with the locally non-geometric fluxes R i, jklm , R ijkl , and R of [81], respectively. The flux R 6,2 also may be related to R ij k and R i .
In this manner, the locally non-geometric fluxes and the mixed-symmetry potentials are in one-to-one correspondence, and moreover, they are associated with the domain-wall branes.
Note that the conjectured electric-magnetic duality relation does not have a manifestly duality symmetric form. It will be an important task to manifest the covariance similar to the approach of [78].

E (6;4) -branes
We can repeat the duality transformations and obtain the following solutions,  is not invariant under the S-duality transformation, but since the R-flux is invariant under the S-duality, the apparent non-invariance is due to a particular gauge choice.
The R-flux, or the magnetic charge of the 1 (2,4,1) 6 -brane is invariant under the S-duality.

Exotic-brane solutions in M-theory
By uplifting the defect-brane solutions in type II theories to M-theory, we obtain the following defect-brane solutions: where S i j 1 ···jp ≡ ∂ i Ω j 1 ···jp and S i j 1 ···j 8 , k ≡ ∂ i Ω j 1 ···j 8 , k . The direction z represents one of the internal ones that is not necessary to be the M-theory direction, which we denote M. If we define the dual field strengths,

Solutions for space-filling branes
We have completed the full list of the "elementary" domain-wall solutions. We can straightforwardly continue the duality rotations to obtain all of the space-filling branes given in Let us perform a formal T -duality along the x 9 -direction to the D8-brane solution (4.13).
We then obtain the D9-brane solution, Similarly, we can obtain the solutions of the 5 4 2 -brane as where τ 1 = mx 8 and τ 2 = h 0 + m |x 9 | . This is also a solution of DFT.
We can easily construct the space-filling solutions, but their interpretation is not clear.
For example, the D9-brane background is expected to be a flat spacetime, but the above solution contains non-trivial winding-coordinate dependence. Recently, a certain limit which removes the winding-coordinate dependence was discussed in [97], where the harmonic function becomes a constant. This may be useful to relate the above solutions to the conventional space-filling solutions such as the D9 solution. It is also not clear how to define the suitable fluxes in these backgrounds. In this paper, we will not address any further issues about the backgrounds of the space-filling branes.

Projection condition for Killing spinors
As it is well known, the supergravity solutions of the standard type II branes admits 32component Majorana-Weyl Killing spinors ε 1 and ε 2 satisfying For each background, the Killing spinors satisfy a certain projection condition that depends on the brane. Here, for convenience, we introduce the Weyl basis which act on ǫ ≡ (ǫ 1 , ǫ 2 ) T . Then, the projection condition for each brane background is expressed as follows (see [111] for a textbook): Under a T -duality transformation along the y-direction, the spinor ε 1 is invariant while ε 2 is transformed as ε 2 → Γ y ε 2 . The S-duality rule has been studied in [112,113] and it mixes ǫ 1 and ǫ 2 as By using these rules, the projection conditions for many exotic branes were studied in [114] (see also [7] for the conditions for the exotic defect-brane backgrounds in M-theory).
Here, we extend the analysis of [114] to all of the "elementary" exotic branes. We will not show the detailed computation, but the result is very simple. The projection condition for the background of an exotic brane that electrically couples to the mixed-symmetry potential E (n) m 1 ···ma 1 , ··· , n 1 ···na s is given by We can easily uplift the type IIA results to M-theory. For a brane that electrically couples to a mixed-symmetry potential E i 1 ···ia 1 ,··· , j 1 ···ja s , the projection rule becomes In this paper, we do not check the Killing spinor equations explicitly, and leave the detailed analysis for future work.

Exotic brane solutions in deformed supergravities
In the previous sections, we have constructed various exotic-brane solutions in DFT/EFT.
Unlike the case of the standard branes or the defect branes, the obtained solutions explicitly depend on the dual winding coordinates. In this section, we explain that the windingcoordinate dependence in the domain-wall solutions can be removed, allowing us to go back to the standard description. The price to pay is the appearing of massive deformations, together with isometry directions in the supergravity theory.
For example, in the literature, the D8-brane background [66] is known as the solution of the massive type IIA supergravity. In this example, the deformation parameter is nothing but the R-R 0-form potential F 0 . Once we include the winding-coordinate dependence of the D8 solution (4.13) into the mass parameter F 0 , the solution (4.13) without the R-R potential "Unknown (6,2,1)" solutions in [6,67], are reproduced.
Despite in this section we provide several examples of deformed supergravities, we leave for a future work the problem of systematically relate exotic branes and gaugings.

Generalized type II supergravity
In order to get a feeling of the deformed supergravity, it is instructive to review the derivation of GSE [61,62] from DFT [63,64] (see also [65] for a derivation from EFT).

Bosonic sector of type II DFT
The equations of motion of the type II DFT are given as where R M N and R are the generalized Ricci tensor/scalar, and K contains the information of H M N , and the energy-momentum tensor E M N is defined in [36,37] (see also for other conventions). If we parameterize the generalized metric H M N in terms of the conventional fields (g mn , B mn ), and remove the dependence on the winding coordinatesx m , the equations of motion of DFT reproduce those of the conventional supergravity.
On the other hand, in order to derive the GSE from DFT, we suppose that the background admits an isometry. In this case, we can choose a set of ten-dimensional coordinates (x m ) = (x i , x z ) such that all fields are independent of x z . Since the SC allows for one more coordinate dependence, let us introduce a linearx z -dependence into the dilaton Here, we have decomposed the dilaton field into two parts, and in the following, we interpret the x i -dependent fields d(x i ) and Φ(x i ) as "physical" dilatons whereas the winding-coordinatedependent part I m is a (non-dynamical) Killing vector. Regarding the R-R field, since the field strength F takes the form Since I m trivially satisfies the Killing properties we are essentially considering a nine-dimensional background.
For the "nine-dimensional" supergravity fields (g mn , B mn , Φ, F), the equations of motion of DFT take the following form: where U m ≡ B mn I n ,F ≡ e B 2 ∧ F , and These are precisely the generalized type II supergravity equations of motion [61,62]. When the winding-coordinate dependence vanishes (i.e. I m = 0), they have the same form as the usual supergravity equations of motion.
In this manner, we can consider a slight modification of the supergravity equations of motion by assuming the existence of an isometry in the doubled space. Since DFT is defined well for arbitrary solutions of the SC, we can systematically determine the modifications of, for example, the gauge transformation and the duality transformation rules (see [61,64] for the I-modified T -duality transformation rule).

Another viewpoint in terms of the Scherk-Schwarz reduction
As discussed in the addendum of [63], the ansatz for the dilaton (6.3) can be understood as the Scherk-Schwarz ansatz in DFT [115-117, 121, 127] (see also [65] where the derivation of GSE from a Scherk-Schwarz compactification of EFT was originally discussed). An ansatz generally introduces gaugings (6.10) which are constrained to satisfy the consistency constraints such as which are closely related to the SC. For the ansatz (6.3) and a constant twist matrix U , we obtain a constant flux f M which satisfies the consistency conditions. According to [65], this corresponds to a nine-dimensional deformed supergravity generated by the gauging of the trombone symmetry and a Cartan subgroup of SL(2) in type IIB [118,119]. In this sense, the introduction of the linear winding-coordinate dependence can be regarded as a systematic way to introduce constant gaugings satisfying the consistency constraints.

5 3 2 solution in a deformed type II supergravity
Let us next consider the 5 3 2 (12345, 678) solution (4.21). In this case, the linear windingcoordinate dependence is contained in the β-field β 67 = mx 8 . For generality, we introduce an arbitrary constant c and decompose the β-field as 7 . (6.12) We then regard the U mn as a part of a β-twist matrix, 13) in the sense of the Scherk-Schwarz ansatz (6.9) (see [120] for a recent study on this twist).
As we infer from their definition, the dual fields associated with the untwisted (or physical) generalized metricĤ M N are 14) or equivalently, in terms of the conventional fields [see (4.17)], Since the winding-coordinate dependence is absorbed into the twist matrix, the solution no longer depends on the winding coordinates. In particular, when we choose c = 0 , the solution is simplified as where the asymmetry between {6, 7} and 8 disappears.
According to the gauged DFT [115-117, 121, 127], the twist matrix changes the NS-NS part of the DFT Lagrangian as 17) whereR is the generalized Ricci scalar forĤ M N and f M N P is a gauging defined as whereÂ 1 (x i ) does not include the x 8 dependence that vanishes in the D8-brane solution (4.13).
As it was studied in [59], in such case, the R-R 0-form field strength becomes constant and the equations of motion of DFT reproduce those of the Romans massive IIA supergravity [58]. The modifications of the gauge symmetry and T -duality transformation rules can be reproduced from those of DFT by considering the ansatz (6.23). Once the winding-coordinate dependence is absorbed into the mass parameter, or the deformation of the supergravity, the D8-brane solution (4.13) without R-R fields becomes a solution of the Romans massive type IIA supergravity. Namely, the windingcoordinate-dependent solution becomes a winding-coordinate-independent solution of the modified supergravity.
As the R-R field depends on a winding coordinate, for the SC to be satisfied, one might expect that it is necessary to require the existence of an isometry direction. However, in this case, as it was shown in [59], by relaxing the strong constraint to the weak constraint, we can formulate the massive IIA supergravity in ten dimensions, rather than 9. This may be understood as follows. We expect that the locally non-geometric R-fluxes will always play the role of the gaugings and the consistency conditions will require conditions like R ···m ∂ m = 0 .
However, in the special case of the D8-brane, the corresponding R-flux (i.e. the R-R 0-form field strength F 0 ) does not have any index and we cannot write a condition for the derivative.
In terms of M-theory, a D8-brane is uplifted to the 8 (1,0) -brane (also known as the KK9Mor the M9-brane) and the associated R-flux is R i,i . Therefore, in this case, we may need to require a condition R i,i ∂ i = 0 . This consideration is consistent with the fact that the eleven-dimensional uplift of the massive IIA supergravity depend on a certain Killing vector in the eleven dimensions [122].

KK8A and M9 solutions
Let us consider the solution of the 7 (1,0) 3 (1 · · · 7, , 8)-brane (5.27), which is also known as the KK8A-brane. At the same time, we consider its eleven-dimensional uplift, the solution of the 8 (1,0) 12 (1234567z, , 8)-brane (5.160). In this case, the linear winding-coordinate dependence is included in γ 8 = mx 8 , or in terms of M-theory, A 8 =G 8M /G MM = −m y 8M . We thus consider the following twist for the generalized metric in EFT: where K i j is a matrix representation of the GL(n) generator given in (B.2). Substituting this ansatz into the EFT action or the equations of motion, we obtain a deformed type IIA supergravity, which will effectively be nine-dimensional due to R m, m (3) ∂ m = 0 . Since the winding-coordinate dependence has been absorbed into the twist matrix, we obtain the following solution of the deformed type IIA supergravity, (6.26) By translating the dual fields into the conventional fields (recall (5.23)), we obtain This is precisely the KK8A solution given in Eq. (6.20) of [6]. By choosing c = 0 , we obtain the KK8A solution originally obtained in [67], which is not a solution of any type II supergravity, but instead of a deformed ten-dimensional supergravity (which is not Romans' supergravity).
On the other hand, the 8 By choosing c = 0 , the conventional metric becomes which is the KK9M solution obtained in [67,123].
On the other hand, if we consider the background of the (p, q)-7(1 · · · 7) brane and perform the T -duality along the x 8 -direction, we obtain a solution corresponding to the bound state of the 8 1 (1 · · · 8)-brane and the 7  (1 · · · 6, 7, 8) solution of (5.27). In this case, the windingcoordinate dependence is contained in γ 78 = mx 8 . The corresponding twist is where the matrix R α mn can be found in (B.13). In this case, the deformed supergravity will effectively be eight dimensional since the 6 (1,1) 3 -brane requires two isometry directions or the R-flux contains two antisymmetric indices.
By absorbing the winding-coordinates into the twist matrix and choosing c = 0 , the solution (5.23) reduces to a purely gravitational solution, This is precisely a solution corresponding to the "Unknown brane (6,2,1)" obtained in Eq. (6.9) of [67].

Acknowledgment
We appreciate useful discussions during the workshop "Geometry, Duality and Strings 2018" at Departamento de Física, Universidad de Murcia. We also would like to thank Shozo

B Parameterizations of the generalized metric in EFT
In this appendix, we review the parameterization of the generalized metric in E n(n) EFT (n ≤ 7) (see [106] and references therein).

B.1 M-theory parameterization
When we consider M-theory, we can parameterize the generalized metric M IJ in terms of the conventional supergravity fields G ij , A i 1 i 2 i 3 , and A i 1 ···i 6 as follows: M IJ = (L T 6 L T 3M L 3 L 6 ) IJ , L 3 = e where d ≡ 11 − n is the dimension of the external space and we have introduced the matrix representations of the E n(n) generators {K k 1 k 2 , R k 1 k 2 k 3 , R k 1 k 2 k 3 , R k 1 ···k 6 , R k 1 ···k 6 } as we can in principle express the dual fields (G ij , Ω i 1 i 2 i 3 , Ω i 1 ···i 6 ) in terms of the conventional fields (G ij , A i 1 i 2 i 3 , A i 1 ···i 6 ).

B.2 Type IIB parameterization
When we consider type IIB theory, we parameterize the generalized metric as Then, Φ, B mn , C m 1 ···m 2n , and D m 1 ···m 6 are the standard dilaton, the B-field, the R-R potentials, and the dual potential of the B-field.
We can also provide the non-geometric parameterization as (B.20) In this paper, we use the fields (g mn ,φ, β mn , γ m 1 ···m 2n , β m 1 ···m 6 ) to describe supergravity solutions of exotic branes in type IIB theory. we can in principle determine the dual fields in terms of the conventional supergravity fields.

C Contents of the p-brane multiplets
In this appendix, we provide a list of branes contained in the p-brane multiplets.

C.1 "Elementary" branes
We first provide a list of "elementary" branes that are connected to the standard branes.