Exotic branes and mixed-symmetry potentials II: duality rules and exceptional $p$-form gauge fields

In U-duality-manifest formulations, supergravity fields are packaged into covariant objects such as the generalized metric and $p$-form fields $\mathcal{A}_p^{I_p}$. While a parameterization of the generalized metric in terms of supergravity fields is known for U-duality groups $E_n$ with $n\leq 8$, a parameterization of $\mathcal{A}_p^{I_p}$ has not been fully determined. In this paper, we propose a systematic method to determine the parameterization of $\mathcal{A}_p^{I_p}$, which necessarily involves mixed-symmetry potentials. We also show how to systematically obtain the T- and S-duality transformation rules of the mixed-symmetry potentials entering the multiplet. As the simplest non-trivial application, we find the parameterization and the duality rules associated with the dual graviton. Additionally, we show that the 1-form field $\mathcal{A}_1^{I_1}$ can be regarded as the generalized graviphoton in the exceptional spacetime.


Introduction
In the previous paper [1], we have conducted a detailed survey of mixed-symmetry potentials in 11D and type II supergravities. By considering their reduction to d dimensions, they yield various p-form fields A Ip p , which transform covariantly under E n U -duality transformation (n = 11 − d). In the U -duality-covariant formulation of supergravity known as exceptional field theory (EFT) [2][3][4][5] (see [6][7][8][9][10][11][12][13][14] for earlier fundamental works), and the U -duality-manifest approaches to brane actions [15][16][17][18], the p-form fields A Ip p play an important role in providing U -duality-covariant descriptions. However, to make contact with the standard descriptions in supergravity and brane actions, explicit parameterizations of A Ip p are needed. In this paper, we propose a systematic method to determine the parameterization of A Ip p by utilizing the equivalence between M-theory (or type IIA theory) and type IIB theory. In our method, in addition to the parameterization of the p-form fields, the duality transformation rules of various potentials can also be obtained. As the first non-trivial example, we obtain the Tand S-duality rules for the dual graviton, equations (2.50)-(2.53) and (2.65), respectively.
In EFT, the fundamental fields are the generalized metric M IJ and p-form fields A Ip p , as well as certain auxiliary fields. For E n EFT with n ≤ 8 , the parameterization of the generalized metric has been determined in [13,19] by means of the bosonic fields in 11D supergravity.
The parameterization in terms of type IIB supergravity has been determined in [20,21] for E n EFT with n ≤ 7 . They are nothing more than the two different parameterizations of the same object M IJ , and as was concretely realized in [22], we can relate the two parameterizations through some redefinitions of fields. As was shown in [22], by rewriting the M-theory fields in terms of type IIA fields, these field redefinitions are precisely the T -duality transformation rule. However, the analysis of [22] is limited to the E n EFT with n ≤ 7 , where the generalized metric contains only the standard p-form potentials. In this paper, we extend their analysis to the case of E 8 EFT, and find the T -duality and S-duality rule for the dual graviton. This gives a non-trivial check of our duality rules for the dual graviton advertised in the first paragraph.
If we look at the explicit parameterization of the 1-form field A I 1 , its first component A i 1 is the graviphoton. In 11D, the graviphoton is defined asÂ i µ ≡ĝ µνĝ νi , by using the 11D inverse metricĝMN and the metricĝ µν in the external spacetime. In this paper, we propose that the 1-form field A I 1 can be regarded as a generalized graviphoton in the exceptional spacetime where MÎĴ (Î = {µ, I}) is the inverse generalized metric in E 11 EFT. We also find that the parameterizations of the higher p-form fields A Ip p (p ≥ 2) can be easily obtained from that of the 1-form A I 1 through a simple antisymmetrization of indices.
2 Parameterization of the 1-form A I 1 In this section, we explain our method to determine the parameterizations of the 1-form A I 1 . The index I transforms in a fundamental representation of the E n algebra with Dynkin label [1, 0, . . . , 0], known as the vector representation or the particle multiplet. Our approach relies on the existence of two equivalent descriptions of EFT by deleting different nodes of the E n Dynkin diagram; M-theory and type IIB theory (see [1] and references therein for details): . (2.1) As it is explained in the accompanying paper [1], in terms of M-theory, the 1-form field A I 1 is decomposed into SL(n) tensors as follows: where i, j = d, . . . , 9, z are indices of the fundamental representation of SL(n). On the other hand, in terms of type IIB theory, the 1-form field is decomposed into SL(n − 1) × SL (2) tensors as follows: where α, β = 1, 2 are the SL(2) S-duality indices and m, n = d, . . . , 9 are indices of the fundamental representation of SL(n − 1). In order to stress the difference between the two parameterizations, we have denoted the 1-form in type IIB parameterization by A I µ . Although we know the tensor structures of each component, it is not obvious how to determine the explicit parameterization in terms of the standard supergravity fields, which is the main subject of this paper.
As demonstrated in [22], the two decompositions (2.2) and (2.3) can be related by using the equivalence between M-theory on T 2 with coordinates (x α ) = (x y , x z ) and type IIB theory on S 1 with a coordinate x y : Type IIA theory/S 1 o o T -duality along x y /x y / / Type IIB theory/S 1 .

(2.4)
Here, x z is a coordinate along the M-theory circle, and the coordinate x y in M-theory (or type IIA theory) is mapped to the coordinate x y in type IIB theory under the T -duality. By using the map, we can rewrite various quantities in M-theory in terms of type IIB supergravity.

Supergravity fields
In order to discuss the parameterization, we will briefly explain the supergravity fields considered in this paper. We basically follow the convention of [22].
When we consider a compactification to d dimensions, the 11D metricĝMN is decomposed as where we have defined the graviphoton asÂ i µ ≡ −ĝ µkĜ ki =ĝ µνĝ νi .
Type IIA supergravity: When we consider type IIA supergravity, we use the following standard 11D-10D map: where we have added the hat to the subscript, likeÂp , to stress that it is a p-form in 11D. In our convention, the dual gravitonÂ8 ,1 = {Â8 ,1 ,Â8 ,z } follows the 11D-10D map, whereÂ8 ,z corresponds toN studied in [23]. The metric and the graviphoton are defined as Then we find the 11D-10D map for the graviphoton, Type IIB supergravity: In type IIB theory, in addition to the standard Einstein-frame metric g M N , we consider the following SL(2) S-duality-covariant tensors,

11)
We also consider the dual graviton A 7,1 , whose behavior under duality transformations is to be determined. Upon compactification to d dimensions, the graviphoton is introduced as

Strategy: Linear map
Here, let us explain the detailed procedure, how to determine the parameterization of the 1-form in both the M-theory and type IIB languages, where ellipses stand for the rest of components that complete the U -duality multiplet which, potentially, involve further mixed-symmetry potentials.
To determine the parameterization, we make the following modest assumptions: • The M-theory fields A 1;p,q,r,··· and the type IIB fields A α 1 ···αs 1;p,q,r,··· are respectively parameterized by the following fields: • The top form is normalized with weight one: A µ;p,q,r,... =Â µp,q,r,... + (sum of products of potentials) , A α 1 ···αs µ;p,q,r,... = A α 1 ···αs µp,q,r,... + (sum of products of potentials) . (2.17) According to these, the first components of the 1-forms should be, respectively, In the following, we explain the procedure to determine the components with higher level, which is based on [22]. In order to utilize the map (2.4), we decompose the physical coordinates on the n-torus in M-theory as (x i ) = (x a , x α ) (a, b = 1, . . . , n − 2) and those on the (n − 1)torus in type IIB theory as (x m ) = (x a , x y ) . Under the decomposition, the 1-form fields (2.14) are decomposed into SL(n − 2) × SL(2) tensors as follows: where toroidal directions (either compactified or T -dualized) are shown explicitly. In terms of the Dynkin diagram given in (2.1), in M-theory we have first performed the level decomposition associated with the node α n . Secondly, we have done the level decomposition associated with α n−2 . On the other hand, in type IIB theory the order is reversed. In the end, we obtain the same decomposition. Indeed, the set of SL(n − 2) × SL(2) tensors appearing in (2.19) have the same structure. Then, we make the following identifications [22]: where we have defined We will refer to the set of linear relations established in (2.20) as the linear map. Actually, by using a constant matrix S I J , it can be rewritten as We note that this identification has been originally proposed in [24] in the context of E 11 . Now, for simplicity, we assume the standard T -duality rule for the NS-NS fields 1 g AB where A, B = {µ, a} (i.e. nine directions except the T -dual direction x y or x y ). From these, we obtain the T -duality rule for the graviphoton By using the 11D-10D relation (2.10), the first rule gives which is nothing but the first row of (2.20).

Detailed procedures
We will continue this process by considering the index structure. The second components of the 1-form in the M-theory and the type IIB parameterization are generically expanded as where c 1 and c 2 are parameters to be determined.
On the other hand, the second rule of (2.24) and the 11D-10D relation (2.10) giveŝ and by comparing this with the α = y component of (2.27) , we find c 2 = 1 .
Similarly, the map A µ;yz On the other hand, the second line of (2.23) and the 11D-10D relation (2.7) givê By substituting the second relation into the left-hand side of (2.29) and using g µy = −(A a µ g ay + A y µ g yy ) , we obtain A a µ g ay + A y µ g yy − c 1 A a µ g ay g yy which shows c 1 = 1 . Thus, the parameterizations of A µ;i 1 i 2 and A α µ;m have determined as In order to determine the parameterization of further components of the 1-forms, the T -duality rules (2.23) are not enough and we need additional T -duality rules. To find the T -duality rules, we assume that • The T -duality rules have the 9D covariance (in the nine directions x A orthogonal to the T -duality direction x y ) .
• The metric appears in the T -duality rule only through the combination g Ay gyy and the graviphoton does not appear explicitly.
By using these assumptions, we obtain the set of standard T -duality rules.
For example, from α = z component of (2.27), we find In terms of the type IIA field, this is equivalent to and by using the identity g µy = −(A a µ g ay + A y µ g yy ) , we obtain From the assumption that the T -duality rule does not contain the graviphoton explicitly, this implies the standard T -duality rule, (2. 36) or conversely, where we have employed the standard rule C y Similarly, if we consider the linear map A µ;aα In particular, for α = y , we obtain a map between the type IIA/IIB fields, and this is equivalent to (2.40) Then, we find the T -duality rule C ABy

Further steps
We can further proceed by considering a general expansion of the SL(2) singlet A µ;m 1 m 2 m 3 ,

Results
By continuing the above procedure, we have determined the M-theory parameterization as Remarkably, the two tensorsN µ;p,q,r,... andN k;p,q,r,... in each row can be regarded as particular components of 11D-covariant tensors: where the meaning of the equivalence ≃ is explained below.
As discussed in [1], for any mixed-symmetry potential, not all of the components couple to supersymmetric branes. For the dual graviton A µ;i 1 ···i 7 ,i , only the components satisfying couples to supersymmetric branes. The components which do not couple to supersymmetric branes correspond to the E 11 roots α satisfying α · α < 2 , and they are not connected to the standard p-form potentials under T -duality and S-duality. Since our procedure to determine the parameterization is based on T -duality and S-duality, it can only provide the parameterization of the components which couple to supersymmetric branes. In this sense, it is more honest to express the last equation of (2.44) aŝ (2.48) The last component N 1;6,1 is relatively long, and the S-duality invariance is not clear. However, this is because of the definition of the dual graviton A 7,1 . As we will see later [in Eq. (3.47)], a certain redefinition of A 7,1 makes the expression of N 1;6,1 simpler.

T -duality rule
In addition to the parameterizations, we have obtained the T -duality rules as follows: For the 6-form potential B 6 and the dual graviton A 7,1 , we find The T -duality rules (2.50) and (2.52) coincide with the known results [25] (see Appendix A therein), for which the following identification of supergravity fields are needed: On the other hand, (2.53) has been obtained in [26], where B 2 = 0 and C 2 = 0 are assumed.
If we truncate B 2 and C 2 , we have A 4 = C 4 and the T -duality rule (2.53) reduces to More explicitly, according to the restriction (2.45), the direction x ≡ B must be contained in In the above computation, we have shown only T -duality transformations from type IIB to type IIA, but we can easily find the inverse map. The standard rules (2.49) have the same form even for the map from type IIA to type IIB, Regarding the 6-form potential and the dual graviton, the results are as follows: Now, let us comment more on the restriction rule. In the T -duality rule (2.53), we are assuming that B is contained in {A 1 , · · · , A 6 } . When the restriction is removed, we expect the right-hand side of the T -duality rule is modified. In general, the components which do not satisfy the restriction is in the same orbit as the (α = β)-component of the type IIB potential A αβ 8 , which is electric-magnetic dual to the 0-form potential m αβ . Therefore, it will be possible that A αβ A 1 ···A 6 By appears on the right-hand side of (2.53).

S-duality rule
The standard S-duality transformation rules are reproduced as follows: (2.64) From the S-duality invariance of A µ;m 1 ···m 6 ,m , we also find

Another approach based on the generalized metric
In this section, we discuss another derivation of the T -/S-duality transformation rule for the dual graviton, which is based on the generalized metric. We also explain another method to determine the parameterization of the 1-form A I µ . In d dimensions, scalar fields are packaged into U -duality-covariant object called the generalized metric, which are denoted as M IJ and M IJ in M-theory and type IIB, respectively.
The generalized vielbein, E I J and E I J respectively, are defined such that According to [13], the generalized vielbein can be constructed as follows. We first consider the positive-root generators of the E n algebra, which are summarized as in the M-theory parameterization and as in the type IIB parameterization. We also consider the Cartan generators, in the M-theory parameterization (D ≡ K i i ) and in the type IIB parameterization (D ≡ K m m ). Then, we prepare the matrix representations of these generators in the vector representation. In the M-theory parameterization, the matrix representations have been obtained in [13] for n ≤ 7 and in [19] for n = 8 . In the type IIB parameterization, they have been determined in [20,21] for n ≤ 7. The results for n = 8 are given in Appendix B. Then, we define the generalized vielbein in the M-theory parameterization as and the generalized vielbein in the type IIB parameterization as (3.7) The objectsÂ and A are again the M-theory and type IIB fields respectively, expressed in a new basis. Namely,Â and A are respectively related toÂ and A by field redefinitions, as we will show in this section. The ellipses in both parameterizations disappear for n ≤ 8 .
The generalized metrics (3.1) are then expressed as where we have defined the untwisted metrics aŝ which are parameterized by the supergravity fields h k (vielbein) and h k (vielbein and m αβ ) .
Explicitly, the untwisted metrics take the following form: On the other hand, the twist matrices L and L contain various gauge potentials, which can be computed by using the matrix representations of the E n generators given in Appendix B.
As we have introduced the parameterization of the generalized metrics, let us explain the procedure to obtain the duality rules, which has been proposed in [22] for n ≤ 7.

Linear map between generalized metrics
Here, we explain how to determine the duality transformation rules from the generalized metric. As we have discussed in Section 2, in the M-theory and type IIB parameterizations, we are using different basis, which are related through the linear map (2.22). Accordingly, the generalized metrics in the two parameterizations are related as The explicit form of S I J has been obtained in [22] only for n ≤ 7 , and in this paper, we extend the result to be applicable to n ≤ 8 . Under the linear map, the generalized coordinates are transformed as (3.14) Since the matrix size of S I J is very large (21 × 21), we show the linear map as follows: with δ j 1 ···jn i 1 ···in ≡ n! δ j 1 ···jn i 1 ···in . The constant matrix S I J can be read off from the above map between the coordinates. We can check that the matrix S I I satisfies the property under the generalized transpose, which is defined for a matrix A = (A I J ) as namely the standard matrix transpose ⊺ followed by a flip in the position of the indices. This property shows that the flat metric is preserved under the linear map, Now, the constant matrix S I J has been completely determined and the relation (3.13) connects the two parameterizations. By comparing both sides, we can express the M-theory fields in terms of the type IIB fields, and vice versa. In the case n ≤ 7 , the generalized metric does not contain the dual graviton, but it appears in n ≥ 8 and here we consider the generalized metric in E 8 EFT.

Connection between two parameterization
By comparing the two parameterizations (3.13) of the E 8 generalized metric, we find the following relation between the M-theory fields and type IIB fields:

27)
A a 1 ···a 6 yz,b for M-theory fields and for type IIB fields, and by using the 11D-10D map, these relations are precisely the T -duality rules obtain in Section 2.5. The S-duality rule for the new dual graviton is simply which is consistent with (2.65) under the identification (3.31).
Note that, in order to obtain the duality rules for the higher mixed-symmetry potentials, we need to consider the E n generalized metric with n ≥ 9 .

1-form A I µ as the generalized graviphoton
In Section 2, we found that the 1-form gauge field A I µ has a simple structure in terms of the tensorsN and N, In fact, this combination has a clear origin. The basic idea is as follows.

(3.36)
This leads us to define the generalized graviphoton as which transforms covariantly under O(n − 1, n − 1) transformations, and is sometimes used in the double sigma model (see for example [18,27]). By using we observe that the generalized graviphoton can be expressed as which has the same structure as (3.33).

Generalized graviphoton in EFT:
We now consider the case of EFT starting with the generalized metric MÎĴ in E 11 EFT. Denoting the inverse matrix of M µν by m µν , we define the generalized graviphoton as In the following, we show that this A I µ is precisely the 1-form considered in Section 2. To this end, let us recall that the generalized metric has the structure By using the fact that the matrix L has a lower-triangular form, we find Then, we obtain In order to show that this is the same as the 1-form considered in Section 2, let us compute the explicit form of (L − ⊺ ) µ I in M-theory/type IIB parameterizations. In the M-theory parameterization, L is defined as (3.6) and by using the matrix representations of the E 11 generators given in Appendix B, we obtain where i ∈ {i 1 , . . . , i 7 } has been assumed for the fourth row. By using the identification (3.29), (L − ⊺ )N I is precisely the same asNN I given in (2.44) and the generalized graviphoton is the same as the 1-form (2.43).
On the other hand, in the type IIB parameterization, L is defined as (3.7) and we obtain where m ∈ {m 1 , . . . , m 6 } has been assumed for the fifth row. Again by using the identification Here, let us comment on the relation to the series of papers [28][29][30]. The standard wave solution in 11D supergravity has the non-vanishing flux associated with the graviphoton A i µ . In [28][29][30], the wave solution was embedded into EFT, which has non-vanishing A I µ . Then, by rotating the duality frames, various brane solutions were obtained in EFT. Particularly in [30], the 1-form A I µ was regarded as the graviphoton in the (4 + 56)-dimensional exceptional space. Since all of their brane solutions in EFT couples to the generalized graviphoton A I µ , branes were interpreted as a kind of generalized waves in the exceptional space. Although the explicit parameterization of A I µ was not determined there, conceptually, their idea is closely related to the result obtained here.

Parameterization of A I p p
In this section, we study the parameterization of the higher p-form fields A Ip p .

2-form
The 2-form gauge field A I 2 2 transforms in the string multiplet, characterized by the Dynkin  label [0, . . . , 0, 1, 0] . It is decomposed as and for example, the first component, in each parameterization, can be expanded aŝ by introducing parameters m 1 , m 2 , b 1 , and b 2 . We already have the T -duality rules, and by following the same procedure as the 1-form, we can determine these parameters.
Repeating the procedure, we find the parameterization (4.5) Interestingly, the tensorsN and N are precisely the same as those defined in Section 2.4. The origin of this simple structure can be understood as follows.
For example, let us consider the map (2.38) in which both sides are connected through T -duality. However, the T -duality rule is 9D covariant, and even if we replace the index a by the 9D index A = (µ, a) , the above relation is still satisfied. Then, choosing A = ν and antisymmetrizing µ and ν , we get which connects the first row of A I 2 µν and the first row of A I 2 µν . In this manner, simply by replacing an internal index a with an external index ν and acting the antisymmetrization, we obtain the parameterization of the 2-form from the result of the 1-form.
In the literature, several components of the 1-form and 2-form have been studied for example in [3,31]. By following the notation of [18], their M-theory parameterization are while the type IIB parameterizations are By comparing, for example A µmn with B µνm , we find that their results also follow the antisymmetrization rule and seem to be consistent with our results up to conventions.

3-form and higher p-form
Similar to the case of the 2-form, a parameterization of a general p-form can be obtained by acting the antisymmetrization to that of the 1-form. In the case of 3-form, we obtain Compared to the 2-form, the first component in type IIB side N α µ;ν has disappeared because the number of indices is not enough to account for a 3-form. The 4-form is In such case, the r-form field is not unique and we cannot fix the parameterization unambiguously.

Summary and Discussion
In this paper, we have proposed a systematic way to determine the parameterization of the p-form field A Ip p . As a demonstration, we have determined how the dual graviton enters the p-form field. We have also determined the duality rules for the dual graviton, which have been partially studied in the literature. Our procedure is based on the (factorized) T -duality and S-duality transformations, which form a subgroup of the full U -duality group. Accordingly, our procedure cannot determine the contribution of the mixed-symmetry potentials which do not couple to any supersymmetric branes. However, we have provided another approach to determine the parameterization of A Ip p . We have found that the 1-form field is precisely the generalized graviphoton A I µ = m µν M νI defined by the E 11 generalized metric. By following the procedure of [6,32,33], we can in principle determine the parameterization of the E 11 generalized metric level by level. We can then determine the full parameterization of the 1-form field. As we have shown, once the parameterization of the 1-form field is determined, we can easily obtain the parameterization of the p-form field by antisymmetrizing the indices.
As future directions, it is interesting to revisit the worldvolume actions of exotic brane. In the case of exotic branes, the Wess-Zumino term contains the mixed-symmetry potentials, but at present, the explicit forms of the brane actions are known for a few examples [26,[34][35][36]. A manifestly U -duality-covariant Wess-Zumino term, which employs the p-form fields A Ip p , has been proposed in [15] and it is important to clarify the connection to the results of [26,[34][35][36] by using the concrete parameterization of A Ip p . It is also interesting to develop another Uduality-manifest approach to brane actions [17,37] (see also [16,18] for a similar approach).
It is also useful to study the duality transformation rules for more mixed-symmetry potentials beyond the dual graviton. By following the procedure proposed in this paper, it is a straightforward task to determine such duality rules. Recently, T -duality manifest formulation for mixed-symmetry potentials has been studied in detail in [38], which aims to be more useful to determine the T -duality rules. Nevertheless, in order to consider the S-duality rule or the M-theory uplifts, our U -duality-based procedure would potentially prove more useful.

A Notation
In this appendix, we summarize the notation that has been used along this work to denote various fields corresponding to each theory and each dimension, as well as the different types of indices.
M-theory and type IIA/IIB theory are defined in D dimensions, where D = 11, 10 respectively. Upon a dimensional reduction on a torus, we have a d-dimensional supergravity theory, with a global symmetry group E n , where n = D − d. According to this, all the splittings of the M-theory and type IIB coordinates and the higher/lower-dimensional indices that have been used are shown in Figure 1. The D-dimensional coordinates in M-theory and type IIB theory are denoted by xM and x M , respectively.
In addition, indices for the p-form multiplet are denoted as I p in M-theory and as I p in type IIB theory. In particular, for the 1-form, we denote I ≡ I 1 and I 1 ≡ I . In type IIB theory, the index of the vector representation of the SL(2) S-duality group is represented by α = 1, 2.
In Table A.1, we summarize the notation that we have used to represent the fields of various theories. Fields transforming as U -duality multiplets are considered. Similarly, standard supergravity fields of M-theory and type II theories, and the lower-dimensional fields that arise after compactification are considered. Field M-theory Type IIB Type IIA

B E n generators
In this appendix, we show the explicit matrix representation of the E n generators in the vector representation. In the M-theory parameterization, our matrices are consistent with [19].
Through the linear map from M-theory parameterization to type IIB parameterization, we find the matrix representations also in the type IIB parameterization, which is new.
Here, we show the results for E 8 , but the E n generators with n ≤ 7 can be easily obtained through a truncation. For example, an E 8 generator R i 1···8 ,i disappears in E 7 because the index i ranges over seven directions and i 1···8 automatically vanishes. Conversely, our E 8 generators can be understood as a truncation of the E 11 generators. In E 11 , the matrix representation becomes infinite dimensional, but the first several blocks are the same as the E 8 generators. Accordingly, although we have computed the matrix (L − ⊺ ) in (3.46) by using the E 8 generators, the first four rows do not change even if we use the matrix representation of the E 11 generators. 3 In that sense, the results given in this appendix can be understood as a truncation of the E 11 generators. 3 To be more precise, in our matrix representations in M-theory, in the fourth row and below that, we have used Schouten-like identities; i.e. terms with antisymmetrized nine indices (· · · ) [i 1 ···i 7 ij] has been dropped because they disappears automatically in n ≤ 8 . However, this does not affect the computation of (L − ⊺ ) in (3.46) because the restriction rule i ∈ {i1, . . . , i7} has been assumed there and terms with the structure (· · · ) [i 1 ···i 7 ij] disappear even for n = 11 . In this sense, (3.46) can be understood as obtained from the E11 generalized metric.

B.1 M-theory parameterization
In the M-theory parameterization, the E n generators are decomposed as where the ellipses disappear for n ≤ 8 . In this appendix, we may use a short-hand notation for the multiple indices, We also using a notation, If we restrict to the case n ≤ 8 , they satisfy the following commutation relations: 4 If we consider the E11 algebra, for example R i 1 i 2 i 3 , Rj 1 ···j 8 ,j needs to be modified as R i 1 i 2 i 3 , Rj 1 ···j 8 ,j = We note that our convention will be related to that of [19] as follows: Now, we show the matrix representations of these generators in the vector representation.
In the M-theory parameterization, the vector representation (for n ≤ 8) is decomposed as where y [i 1···7 ,i] = 0 . In this paper, in order to reduce the matrix size, we have combined y i 1···7 ,i and y i 1···8 , and our y i 1···7 ,i do not satisfy y [i 1···7 ,i] = 0 . We then find that the following matrices (Tα) I J satisfy the above E 8 algebra: qr 1···7 δ l 123 k 123 i 1···7 r δ rs 12 p 123 δ l 123  The set of the positive/negative root generators can be obtained by taking commutators of the simple-root generators E k /F k , and they can be summarized as (B.36)

B.2 Type IIB parameterization
We can transform the E n generators of the M-theory parameterization into the type IIB parameterization by using the linear map (3.14). Namely, we act the following operation to the matrix representations of the generators Then, T (Tα) I J is the matrix representations in the type IIB parameterization. The explicit form of the constant matrix S I J has been determined such that the algebra of the type IIB generators is closed. We also change the name of the generators such that the SL(7) × SL (2) symmetry is manifest. Concretely, we convert the non-positive-level generators of the M-theory parameterization into those of the type IIB parameterization as follows: T (R a 1 ···a 6 yz,a ) R a 1 ···a 6 y,a We then obtain the E n generators (n ≤ 8) in the type IIB parameterization, By using the notations, δ n 1 ···nn m 1 ···mn ≡ n! δ n 1 ···nn m 1 ···mn , their matrix representations are found as follows: (R γδ ) α 12 β 12 ≡ δ α 1 α 2 β 1 (γ ǫ δ)β 2 + δ α 1 α 2 β 2 (γ ǫ δ)β 1 , (B.43)