Exotic branes and mixed-symmetry potentials I: predictions from $E_{11}$ symmetry

Type II string theory or M-theory contains a broad spectrum of gauge potentials. In addition to the standard $p$-form potentials, various mixed-symmetry potentials have been predicted, which may couple to exotic branes with non-standard tensions. Together with $p$-forms, mixed-symmetry potentials turn out to be essential to build the multiplets of the U-duality symmetry in each dimension. In this paper, we systematically determine the set of mixed-symmetry potentials and exotic branes on the basis of the $E_{11}$ conjecture. We also study the decompositions of U-duality multiplets into T-duality multiplets and determine which mixed-symmetry tensors are contained in each of the U-/T-duality multiplets.


Introduction and summary
Maximal supergravities in 10 and 11 dimensions have the E n U -duality symmetry when they are toroidally compactified to d (= 11 − n) dimensions. The E n symmetry can be clearly seen by packaging all of the d-dimensional scalar fields into the generalized metric M IJ , which parameterizes the coset space E n /K n (K n : the maximal compact subgroup of E n ). A coset representative g can be parameterized by using the positive Borel subalgebra of the E n algebra.
For example, in d = 4 (or n = 7), the Borel subalgebra is spanned by K i j (i < j), R i 1 i 2 i 3 , R i 1 ···i 6 (i = 1, . . . , n) , (1.1) where multiple indices are antisymmetric. Indeed, using the scalar fields {Ĝ ij ,Â i 1 i 2 i 3 ,Â i 1 ···i 6 } resulting from the compactification, we can parameterize the coset representative as g = e i<jĥ i j K i j e Here,ĥ i j is the logarithm of the vielbein associated withĜ ij andÂ i 1 i 2 i 3 andÂ i 1 ···i 6 are 3-form and 6-form potentials in 11D supergravity. Then, we can obtain the generalized metric M IJ from a matrix representation of g . However, when we consider the case d = 3 , we find that the standard scalar fields are not enough to parameterize E 8 /K 8 . As pointed out in [1], an additional generator R i 1 ···i 8 ,i (satisfying R [i 1 ···i 8 ,i] = 0) appears in the Borel subalgebra. Then, we need to introduce a non-standard field, namely a mixed-symmetry potentialÂ i 1 ···i 8 ,i (or simplyÂ 8,1 ), which is called the dual graviton.
Mixed-symmetry potentials are indispensable for realizing the duality symmetry manifest.
In addition to the scalars M IJ , p-form fields are also forming some U -duality multiplets.
In order to construct the U -duality multiplets for higher p-form fields, we need to introduce more non-standard potentials. In general, the new potentials can be decomposed into some irreducible representations of SL(n) characterized by Young tableaux. Each of the potentials is denoted asÂ p,q,r,··· when the indices have the following Young-tableau symmetry: (1.3) These objects are generally called mixed-symmetry potentials. Of course, mixed-symmetry potentials are not new independent fields. They are related to the standard supergravity fields through the electric-magnetic duality, much like the 3-form potentialÂ 3 is related to the 6-form potentialÂ 6 . For example, the dual gravitonÂ 8,1 is dual to the graviphoton.
In type II theories, the notation slightly differs. When a d-dimensional p-brane has tension T p = g −n s l s (2πl s ) p R m 1 · · · R m b−p l b−p s R n 1 · · · R nc 2 l c 2 s 2 · · · R q 1 · · · R qc s l cs s s l s : string length, g s : string coupling constant , the object in 10D is called an (exotic) b (cs,...,c 2 ) n -brane. In particular, worldvolume actions for the exotic 5 2 2 -brane and 5 2 3 -brane are studied in [23,25], and they will couple to potentials with the tensor structure A 8,2 . The correspondence is the same as in the case of M-theory, b (cs,...,c 2 ) n ↔ Z b+c 2 +···+cs,c 2 +···+cs,··· ,c s−1 +cs,cs ↔ A 1+b+c 2 +···+cs,c 2 +···+cs,··· ,c s−1 +cs,cs . (1.8) By following the notation of [13][14][15][16], we sometimes denote the potential A as B, C, D, · · · depending on the integer n appended to the name of exotic branes: (1.9) Then, for example, the 5 2 2 -brane and the 5 2 3 -brane couple to D 8,2 and E 8,3 , respectively. Given the existence of a variety of mixed-symmetry potentials/exotic branes in M-theory or type II theories, a natural question is what kind of potentials/branes exist in 11D or 10D.
As discussed in [55], Table 1.2 is consistent with the results of gauged supergravities in various dimensions. 1 Indeed, in the language of the embedding tensor formalism [59,60], the (d − 2)-form and the (d − 1)-form potentials are respectively dual to the scalar fields and the embedding tensors (or deformation parameters) while the d-forms correspond to the quadratic constraints (which ensure the consistent deformations of the theories), and their numbers are the same as those given in Table 1.2. 2 In this sense, the E 11 conjecture is consistent with the requirements of supersymmetry. 3 In addition to the mixed-symmetry potentials, the E 11 conjecture also predicts a variety of brane charges. In [54,63], central charges for all of the branes were identified with weight vectors in the vector representation of E 11 . Despite this representation is also infinitedimensional, by introducing a parameter called level ℓ [64], the charges can be ordered. The 1 See [58] for a recent review of gauged supergravity. 2 This agreement occurs upon discarding the gaugings of the trombone symmetry, which would imply additional deformation parameters and quadratic constraints. We also note that the agreement between the d-forms and the quadratic constraints is not perfect: 248 in d = p = 3 has not been contained in the representation of the quadratic constraint in d = 3 .
Formal aspects of the U -duality-covariant expressions have been studied well, but in order to make the physical picture more transparent, it is useful to get back to the standard description in which the manifest duality symmetry is broken. For example, U -duality-covariant description for the Wess-Zumino term has been studied in [72]. The brane action is described by the U -duality-covariant p-form fields A Ip p , where I p is transforming in the p-form multiplets described in Table 1.2. However, in order to reproduce the standard brane actions, we need to decompose the covariant objects A Ip p into the standard potentials and the mixed-symmetry potentials. While we will postpone the parameterizations to the subsequent paper [73], here we will rather concentrate on more algebraic aspects based on the E 11 conjecture: what kind of mixed-symmetry potential exist in M-theory/type II theory, and which potential is a member of each U -or T -duality representation?
Using the high predictability of the E 11 conjecture, we can find new mixed-symmetry potentials and exotic branes as long as we carry out the corresponding computation. Because of the so many previous studies, it is convenient to clarify both the current status of this topic and the new results appearing in this paper.
Type IIB potentials: When we discuss type IIB theory, it is useful to introduce another level ℓ 9 [76]. The low-level potentials are determined in [76] up to level ℓ 9 = 7 (and some of level ℓ 9 = 8), which is consistent with an earlier work [77]. We have continued the analysis by using SimpLie, and potentials up to level ℓ 9 = 14 have been determined (see Table 2.2).
However, this is still not enough to complete Therefore, the highest level l 9 = 22 corresponds to the potential A [11] 10,7,7,7,7,6 . We note that some potentials in Table 1.3 have some totally-symmetric upper indices α 1 ···αs (α = 1, 2) . They denote that the potential transforms as an SL (2)   (1.14) Then, for a generator A α 1 ···αs p,q,··· with 2 ℓ 9 lower indices, information of the square bracket can be easily determined as A , so we will omit the square bracket in the following.
Moreover, this simple rule (1.14) readily leads to the following one: ✓ ✒ ✏ ✑ a type IIB potential with even/odd level ℓ 9 transforms in an odd/even-dimensional representation under S-duality.
(1.15) For example, a potential with odd level ℓ 9 (whose number of indices is 2 mod 4) cannot be an S-duality singlet.
Type IIA potentials: As usual, type IIA fields can be obtained from M-theory potentials by making the (10+1) decomposition, where the M-theory direction is denoted as x z . In other words, type IIA potentials are characterized by two levels ℓ (the same one as in M-theory) and n (the one appearing in the tension formula). If indices of an M-theory potential is restricted to 10D in type IIA theory, the integer n becomes n = ℓ . On the other hand, n decrease as the number of the M-theory direction z is increased, and we have n ≤ ℓ . In the type IIA case, the relation between the levels and the number of indices of potentials is [13] ✞ ✝ ☎ ✆ 2 ℓ + n = (# of lower indices) − (# of upper indices) .
(1. 16) In [1,76], the set of potentials were determined up to level ℓ ≤ 5 . By decomposing the Mtheory potentials, at least, we can determine the potentials with ℓ ≤ 10 . In order to reproduce where the first integer p in the subscripts denotes that the potential is a p-form in the external writing down a manifestly T -duality-symmetric brane actions. For example in [35], covariant 5-brane actions in d = 6 and d = 8 are (partially) written down by using the potentials of the form D 6;··· (see a recent paper [40] for further details). In order to know what kind of 5-brane is described by that action, we need to identify the detailed constituents of the potentials D.
As detailed in [13], they are made of the dimensional reductions of the familiar potential D 6 that couples to the NS5-brane, and the dual graviton D 7,1 that couples to the KKM, and D 8,2 that couples to the exotic 5 2 2 -brane, etc.,  Brane actions for such exotic branes are discussed in [40]. The O(D, D) potentials with higher level (n ≥ 4) also have not been fully identified, but the ones that may couple to supersymmetric branes are fully determined in [16] up to n = 6 .
In this paper, we make a full list of the O(D, D) potentials in dimensions 3 ≤ d ≤ 9 .
When we consider a compactification to d dimension (3 ≤ d ≤ 9), all of the p-form potentials (which form some U -duality multiplets) are already given in Table 1 The detailed results are given in Appendix G. There, we further decompose the obtained O(D, D) potentials into d-dimensional mixed-symmetry potentials in type IIA/IIB theory.
The E 11 conjecture claims that E 11 symmetry is a true 11-dimensional symmetry even before the compactification, and it contains the O(10, 10) "T -duality" symmetry as a subgroup.
Indeed, in the T -duality manifest supergravity, called double field theory (DFT), the supergravity action has a formal O(10, 10) symmetry. Therefore, the mixed-symmetry potentials in type IIA theory and type IIB theory predicted by the E 11 conjecture can be embedded into O(10, 10)-covariant tensors (see [78] for the original discussion of T -duality in the context of Here, we summarize some observations associated with O(10, 10) tensors and mixedsymmetry potentials: • When we decompose a level-n O(10, 10) tensor into the mixed-symmetry potentials in type IIA or type IIB theory, a symmetry in the index structure arises. For example, an O(10, 10) spinor Cȧ is decomposed into C 1 , . . . , C 9 in type IIA theory, and there is the familiar duality between C 1 ↔ C 9 , C 3 ↔ C 7 , C 4 ↔ C 6 . The sum of the indices on each pair is 10 in this case. In fact, this symmetry extends to potentials with higher level n . 5 Here, we consider d = 0 (rather than d = 10) and the whole physical space is treated as the internal space.
In general, we find the following rule: and D (with n = 2). One of the D 10 , which corresponds to the singlet D, is self-dual, and the summation of indices gives 10 × 2 . In the multiplet D A 1 ···A 4 , the highest weight corresponds to D 6 and its dual is D 10,4 . The total number of the indices is again 10 × 2 .
This kind of duality has been noted in [6,9] in the context of the duality-invariant mass squared in U -duality multiplets.
A stronger rule is as follows (note that A p,...,q,0,...,0 ≡ A p,...,q ): Indeed, when n is odd, the O(10, 10) tensors have a spinor index a orȧ and their decompositions into type IIA and IIB yield different SL (10) tensors. This pattern has been noted in [79] and the odd-n sectors are called the generalized R-R sector. On the other hand, regarding the even-n sectors, some O(10, 10) tensors give the common tensors in both type IIA and type IIB theory (the generalized NS-NS sector) and some correspond to the generalized R-R sector. 6 For an O(10, 10) potential in the (generalized) NS-NS sector, we find the following rule: The highest-weight state in an arbitrary O(10, 10) multiplet (in the NS-NS sector) corresponds to a mixed-symmetry potential with n/2 columns, A m 1 ,...,m n/2 .
(1.22) 6 In a recent paper [80], it has been noted that potentials with n = 2 mod 4 are only in the NS-NS sector while those with n = 0 mod 4 are in the NS-NS sector or the R-R sector. However, this pattern is broken in general. For example, the S-dual of F10,10,7,1 , namely, L10,10,7,1 (n = 10) is contained in the R-R sector.
As we have concretely checked, this rule for all possible O(10, 10) tensors up to level n = 26, we conjecture this is a universal property. In general, the potential A m 1 ,...,m n/2 corresponding to the highest-weight has the smallest number of indices, and from the rule (1.20), its dual potential A 10,...,10,10−m n/2 ,...,10−m 1 has the largest number of indices in that O(10, 10) multiplet.
• There is an additional pattern for the underlined potentials (i.e. the ones that may couple to a supersymmetric brane). If we want to know whether two potentials belong to the same O(10, 10) multiplet, the following rule for n ≥ 2 applies: For example, let us consider potentials H 10, 10,9,7,4 and H 9,9,7,3,2 (with n = 6). Since the integers are 9 − 7 = 2 and 7 − 3 = 4 , they are not in the same O(10, 10) multiplet.
If the integers are the same, in general, we cannot conclude that these potentials are in the same O(10, 10) multiplet. At least up to level n = 7 , we have checked that all of the level-n O(10, 10) tensors predicted by E 11 have different values, meaning that if the above integer is the same, the mixed-symmetry potentials are in the same O(10, 10) multiplet. Even for n > 7 , for all of the mixed-symmetry potentials given in Table 1.4, the integers are in one-to-one correspondence with the O(10, 10) multiplets.
In fact, this rule is equivalent to the following rule [37] under the correspondence (1. (1.24) In particular, the fact that all of the solitonic (i.e. n = 2) branes are five branes 5 m 2 (note that c 1 = b) can be understood from this rule.
• Regarding the S-duality, the rule is the following: Under an S-duality transformation, a level-n potential A p,q,r,··· where 2 ℓ 9 = p + q + r + · · · .
(1. 25) In particular, a potential with n = ℓ 9 /2 is a singlet. For an underlined potential, we find p,q,r,··· is a member of A α 1 ···αs p,q,r,··· where s = |2 n − ℓ 9 | and 2 ℓ 9 = p + q + r + · · · .  representations which contain a supersymmetric brane are marked in red. We note that the domain-wall branes are in one-to-one correspondence with the embedding tensors in the presence of the trombone gaugings [81]. The space-filling branes also correspond to the quadratic constraints in the presence of the trombone gaugings [81] (an exception is 1 in d = 4).
Central charges: In addition to mixed-symmetry potentials, we have also made a list of the central charges. As already explained, low-level charges (1.11) have been identified in the literature. A more detailed analysis was worked out in [9], and U -duality multiples and tensions of d-dimensional p-branes were studied for d ≥ 3 . There, the number of p-branes has been counted and a

Structure
The structure of this paper is as follows. Section 2 consists of the derivations of the mixedsymmetry potentials predicted from the E 11 conjecture and their uplifts into O(D, D) tensors.
Firstly, we briefly review the main aspects of E 11 and the method to obtain the spectra of potentials from M-theory. Secondly, we apply the E 11 conjecture to type IIB theory. We demonstrate that the obtained type IIB potentials are more than enough to reproduce Table   1 In Section 3, we study a set of central charges in M/IIB theory by searching the vector representation of E 11 . We also study their reductions to d dimensions, and explain how to determine the central charges of p-branes from the E 11 conjecture. The main results of Section 3 are summarized in Table 1.5 and Appendices B and C.
In Section 4. we discuss our results and comment some prospects. in Appendix H, we obtain a full list of type II potentials that couple to supersymmetric branes up to level n = 36 . Since the result is too long, we explicitly show the results up to level n = 11 , and the results up to level n = 24 (potential Z) is given in the Ancillary files on arXiv.
The results for higher levels can be generated by using the mathematica notebook, which is also included in the Ancillary files.
2 Mixed-symmetry potentials from the E 11 conjecture In this section, we review the basic idea and the results of the E 11 conjecture. We then provide a detailed survey of the mixed-symmetry potentials. We also study the O(D, D) multiplets of gauge potentials and show how the mixed-symmetry potentials in type IIA/IIB theory are embedded in each O(D, D) multiplet.

Mixed-symmetry potentials and the E 11 generators
The Kac-Moody algebra E 11 , or the very extended E 8 algebra E +++ . (2.1) We denote the Cartan matrix as we can uniquely determine the algebra for the infinitely many E 11 generators. An important property of E 11 is that, if we split the Dynkin diagram by deleting one of the nodes α i (3 ≤ i ≤ 11) , the E 11 decomposes into products of finite dimensional algebras. By using this property, it is useful to introduce an ordering, called "level" [64] (see also [82]).
Here, we consider the level decomposition by deleting the node α 11 , which gives the SL (11) subalgebra associated with simple roots α i (i = 1, . . . , 10) . As usual, for the SL(11) algebra, we introduce the metric and introduce the fundamental weights as By using the SL(11) weights λ i (and recalling α 8 · α 11 = −1), the root α 11 of E 11 can be expressed as where x is a certain vector orthogonal to λ i and α i ( we obtain x · x = − 2 11 . Then, a general positive/negative root α of E 11 is expressed as where m i are non-negative integers and ℓ is a non-negative/non-positive integer. This integer ℓ is called the level associated with the node α 11 . The generators that are obtained by taking commutators with the simple roots E i , F i (i = 1, . . . , 10), and the Cartan generators H i have the level ℓ = 0 . They are nothing but the SL (11) generators and an SL(11) singlet H 11 , which are combined to give the GL(11) generators K i j .
By taking a commutator with E 11 (or F 11 ), the level is increased (or decreased) by 1, and we obtain a set of generators with ℓ = ±1 . It is important to note that, since the generators with a fixed level ℓ form some finite-dimensional representation of SL(11) , we can always decompose the E 11 generators at each level into some irreducible representations of SL (11) .
corresponds to 1760 of SL (11). We can decompose it into representations of SL(4) × SL(7) as , (2.18) where the indices before the semicolon represent the external SL(4) representation while those after the semicolon represent the internal SL(7) representation. Usually, we do not consider the mixed-symmetry potentials in the external space because their interpretation is not clear.
We thus consider only the p-form fields appearing in the first line. All of the other mixedsymmetry potentials in 11D also can be similarly decomposed. As worked out in [55], the number of p-form fields in d dimensions coming from the mixed-symmetry potentials can be summarized as in Table 1.2.
In fact, in order to obtain Table 1.2, it is more efficient to delete a node associated with α d instead of α 11 : ( /) .* -+ , . in [56]. By collecting the p-form representations of the SL(d) subalgebra (with Dynkin index 1st 0 ]), the number of the p-form fields predicted by the E 11 conjecture was determined, even for d = 3 and p = 3 as in Table 1.2.
By further deleting the node α 11 , we can decompose the E n representations into SL(n) tensors. The details are summarized in Appendix G.

Treatment of type IIB theory
In the previous subsection, we obtained mixed-symmetry tensors in M-theory by decomposing the adjoint representation of E 11 by means of the level ℓ associated with the node α 11 . On the other hand, we can obtain the mixed-symmetry tensors in type IIB theory by deleting the node α 9 . In this case, the adjoint representation of E 11 is decomposed into representations of SL(10) × SL(2) , where the SL(2) is the standard SL(2) S-duality symmetry in type IIB theory. We denote the level associated with the deleted node as ℓ 9 , and the potentials with level 1 ≤ ℓ 9 ≤ 14 are summarized in Table 2.2. There, α, β = 1, 2 are SL(2) indices and the multiple SL (2) indices α 1 · · · α s are totally symmetrized. The underlined potentials corresponds to E 11 roots α satisfying α · α = 2 , which means that the potentials may couple to supersymmetric branes, as discussed in the next subsection.

Comments on supersymmetric branes
Here, for completeness, we make some comments on supersymmetric branes. The R-R potentials couple to D-branes, and similarly, standard p-form fields couple to certain supersymmetric branes. However, as discussed in [12,72,83,84], not all of the potentials couple to supersymmetric branes. For example, let us consider the SL(2) S-duality triplet of 8-forms (α, β = 1, 2) in type IIB theory. They are contained in some U -duality multiplets in Table 1.2 (e.g. 3 in d = 9 and p = 8). Two components, A are known to couple to supersymmetric branes, the D7-brane (7 1 -brane) and the NS7-brane (7 3 -brane), respectively. However, we cannot write down the standard Wess-Zumino term for the remaining component A in a gauge-invariant manner [83], and it has been considered that there is no supersymmetric brane that corresponds to this potential. Another example consists of the 10-forms A (αβγ) 9 and A α 9 in type IIB supergravity, which are predicted in [85][86][87] and are shown to be consistent with the supersymmetry algebra. They are also predicted by the E 11 conjecture [79]; for example in d = 9 and p = 9 of Table 1 couple to supersymmetric branes, D9-brane (9 1 -brane) and the 9 4 -brane, but gauge-invariant Wess-Zumino term for the remaining two components cannot be written down. Also from a discussion based on the supersymmetry of brane actions [88], it has been concluded that the two 9-forms do not couple to any supersymmetric branes.
A criterion based on the E 11 algebra has been discussed in [12] to elucidate whether a potential couples to a supersymmetric brane: A root α of the E 11 algebra is associated with a supersymmetric brane if it has the length squared α 2 = 2 . Otherwise (i.e. α 2 = 0, −2, −4, · · · ) it does not couple to any supersymmetric brane. For any root α which corresponds to the standard potential (coupling to a supersymmetric brane), α 2 = 2 is indeed satisfied. Moreover, the T -duality and S-duality (which correspond to the Weyl reflections) do not change the norm α 2 , and branes in the Weyl orbits of the standard branes always correspond to roots with On the other hand, E 11 weights associated with potentials which may not couple to a supersymmetric brane (such as A 8 ) satisfy α 2 < 2 . In Table 1.2, all representations whose highest weights correspond to E 11 roots satisfying α 2 = 2 are colored in red. The uncolored representation do not contain any root with α 2 = 2 . If we compare Tables 1.2 and 1.5, we find that the same colored representations are contained in both tables. This means that all of the supersymmetric branes have the corresponding potential as usual. On the other hand, the physical meaning of uncolored representation is less clear and is not addressed in this paper.
Inside the colored representation, the number of E 11 roots satisfying α 2 = 2 (i.e. the number of supersymmetric branes) was counted in [12] (see also [37]), and the result is summarized in Table 2.3. For p-branes with p ≤ d−4 , all E 11 roots contained in each E n representation satisfy α 2 = 2 , which means that all weights in that E n multiplet correspond to supersymmetric branes.
The situation changes for p-branes with p = d − 3 , which are known as defect branes.
8 For A α 1···s p,q,r,··· , this condition requires that all of the SL(2) indices α1, . . . , αs are the same. By using the map between M-theory and type IIB theory [73,78,89], this rule can be derived from the M-theory rule (2.27).
We also comment on the behavior of the restricted components of the underlined potentials under T -duality transformations. Under a T -duality along the x y direction, the radii R i and the string coupling constant transform as 29) and the exotic b has been noted in [47]. For example, let us consider a potential H 10,7,7,5,1 that is predicted by E 11 . Depending on the choice of the integer i, we obtain the following potentials: All of the potentials on the right-hand side are indeed predicted by E 11 and they are in the same T -duality multiplet. This rule is simple and useful to know the T -duality rule for mixed-symmetry potentials.

O(D, D) multiplets of the mixed-symmetry potentials
We can consider another interesting level decomposition by deleting the node α 10 : ( /) .* -+ , . (2.34) We denote the associated level by n and, from the tension formula discussed in [9], this level n corresponds to the power T ∝ g −n s of the associated brane. Under this level decomposition, the E 11 generators are decomposed into representations of the O(10, 10) T -duality group. Again by using SimpLie, we can execute the level decomposition. The result up to level n = 4 is given in Table 2.4, and there, further decompositions into the type IIA/IIB tensors are made.
We can proceed further the potentials with level n = 5 , and obtained O(10, 10) tensors, The On the other hand, the O(10, 10) tensors which do not contribute to Table 1.2 are as follows: Generalized NS-NS sector:
The resulting potentials with level n = 1, 2 in any dimensions can be summarized as [13,15,16] They are obviously coming from the O(10, 10) tensors B M N , C a . The results for n = 2, 3, 4 are summarized in Table 2.6. As found in [13], potentials with level n = 3 (in 2 ≤ d) can be nicely summarized as 9 (2.41) All of them arise from the reduction of the 10D potential D A 1···4 and D . For level n = 3, as found in [15], the O(D, D) tensors (in 2 ≤ d) can be summarized as Again, they are arising from the 10D potentials E A 12ȧ and Eȧ .
For the higher level potentials, the pattern is much more non-trivial. In 10D, potentials with level n = 4, 5 are     The general pattern for their reduction is non-trivial, but if restricted to the underlined tensors, we can observe the following pattern [16]: (2.44) The p-form multiplets for higher levels n = 6, 7, 8 are summarized in Table 2  of our concern d ≥ 3 , higher-level potentials with n = 9, 10, 11 appear only in the 3-form multiplet in d = 3 , . . For example, for a potential F ([n/2] = 2), we find a series . (2.48) The non-underlined potentials also follow these rules, but we do not find any patterns for their multiplicities. The underlined potentials always have multiplicity 1.
We also studied the decompositions of the E n multiplets into O(D, D) representations.
The results are detailed in Appendix G.

BPS branes and the vector representation
We also consider the vector representation of E 11 , which contains the momenta P i and the brane central charges. As discussed in [54], the multiplet is called the l 1 representation because the highest-weight state corresponds to the fundamental weight l 1 . Here, the fundamental weights l i of the E 11 algebra are expressed as by using the fundamental weights λ i of the subalgebra SL (11) . The l 1 is the highest weight of the SL(11) multiplet [1, 0, . . . , 0] , which corresponds to a tensor P i 1 ···i 10 . Through the dualization, this tensor corresponds to the momenta P i . By taking commutators with the positive generators we obtain infinitely many central charges. The commutation relations for low-level generators have been determined in [54,63,90] as When we take the commutator with E 11 = R 89z , the level ℓ is increased, and at the same time, the number of upper indices is increased by three. Thus, the level ℓ satisfies  .

(3.4)
We then consider the E ++++ 8 generators with level m * = 1 (associated with the node α * ), and we find that these E ++++ 8 generators precisely correspond to the weights in the vector representation of E 11 [63,90]. The low-level central charges up to level ℓ = 7 are given in roots α satisfying α · α = 2 are associated with the supersymmetric branes, and they are underlined. In order to find the central charges in type IIB theory, we delete the node α 9 instead of α 11 , and the result up to level ℓ 9 = 10 is given in Table 3.1.
As it is known well, the momenta and brane charges are mixed under U -duality transformations, and this vector representation is the multiplet for such brane charges. The standard coordinates x i are canonical conjugate to the momenta P i , and for the manifest U -duality covariance, it is natural to introduce additional coordinates y i 1 i 2 , y i 1 ···i 5 , · · · which are conjugate to the brane central charges Z i 1 i 2 , Z i 1 ···i 5 , · · · . This extension of geometry has been discussed in [54,63,90] (see for example [67,68,71,91,92] for further discussion) and this leads to the recent developments in the U -duality manifest formulation of supergravity, known as the exceptional field theory (EFT).

Central charges in d dimensions
When we discuss the standard E n U -duality n ≤ 8 , the spacetime is decomposed into the external d-dimensional spacetime and the internal n-dimensional space. Accordingly, central charges in the vector representations are decomposed into SL(d)×SL(n) tensors. For example, in d = 9 , since the internal space is two-dimensional, only the following charges can appear Z 7,1 → Z 5;2,1 (2), Z 6;1,1 (3), Z 6;2 (1), Z 7;1 (2), Z 9,1,1 → Z 7;2,1,1 (3), Z 8;1,1,1 (4), Z 8;2,1 (2), . (3.6) The same analysis was already done in [9], but the multiplicities are not completely determined. We here determine the multiplicities as well, and the resulting p-brane charges in each dimension d are given in Table 1 Central charges in type IIB theory up to level ℓ 9 = 11 αβ , 5 Z 9,6,4 , Z 9,7,1,1,1 , 3 Z 9,7,2,1 αβ , 7 Z 9,7,2,1 , 7 Z 9,7,3 αβ , 10 Z 9,7,3 , 5 Z In Table 1.5, the E n multiples of the central charges are described, but in terms of the Mtheory or type II theories, the E n multiples can be decomposed into mixed-symmetry tensors in 11D/10D. The decompositions into brane charges in M-theory and type IIB theory are studied in detail in Appendix B and C, respectively. We can see that the momenta P i appear only in the particle (or 0-brane) multiplet, and in the E n EFT, we introduce generalized coordinates x I that are canonical conjugate to the central charges in the particle multiplet.
For example, in the E 8 EFT, we introduce 248 generalized coordinates x I in addition to the 3 external coordinates x µ .

Conclusions
In this work, we have conducted a detailed survey of mixed-symmetry potentials and brane charges in M-theory and type II theories that are predicted from the E 11 conjecture. We also considered their dimensional reductions to d dimensions (3 ≤ d ≤ 9) and checked that all the p-form potentials or p-brane charges form U -duality multiplets. We have also given In this paper, we studied only the spectra of the gauge potentials, but in order to clarify the role of such objects in string/M-theory, it is important to study their dynamics. As these potentials do not appear explicitly in the standard formulations, it is useful to employ the duality-symmetric theories, such as DFT and EFT. Generically, formulations of EFTs in an arbitrary dimension contain p-form potentials A Ip p . Here, under U -duality transformations, the index I p transforms in the p-form multiplet given in Table 1.2. Then, regardless of the duality frame in which the section condition is solved, we should be able to parameterize such p-form fields A Ip p in terms of certain potentials. As detailed in this paper, we already know what potentials can enter in the p-form multiplet A Ip p , but the explicit parameterization (which depends on the convention) still needs to be specified. In the companion paper [73], explicit examples of parameterizations will be given. Once the parameterization is fixed, we can evaluate the supergravity action for the mixed-symmetry potentials as well as the standard supergravity fields. Besides supergravities, we can also study brane actions in a U -duality covariant manner. The p-form fields A Ip p enter also in such theories, and it is important to study the role of mixed-symmetry potentials in brane actions.
If the parameterization of the A Ip p fields is given in terms of M-theory potentials or type II potentials, we can determine the duality transformation rules under T -and S-duality. As discussed in [37], duality rules for mixed-symmetry potentials are very useful to generate new supergravity solutions, and they also will be studied in the forthcoming work [73].

A Convention
In this paper, we employ the convention for the Dynkin diagram of E 11 given in (2.1).

B M-theory branes: E n → SL(n)
In this appendix, we find which brane charges in M-theory that are contained in each E n Uduality multiplet of Table 1.5. The first integer of the central charge Z p;q,r,··· represents that the brane is a p-brane. The remaining integers q, r, · · · denote the type of the SL(n) tensor.
M-theory: d = 9 A 10,1,1 A 8;2,1,1 (3) A 9;1,1,1 (4), M-theory: d = 7    In d = 3 , the task is rather complicated. The 1-form and 2-form gauge potentials are determined in [55] but the details of the 3-form have not been given in the literature. As mentioned in the introduction, in order to reproduce all of the potentials in the 3-form multiplets, we need to determine the E 11 generators up to level ℓ = 17 but it is not easy even by using SimpLie. On the other hand, if we consider the level decomposition of [56] by deleting the node α 3 , we can find that the 3-forms are in the 147250 + 3875 + 248 representations of E 8 .
By using SimpLie, we can first determine a set of the allowed O(10, 10) potentials, whose multiplicities are not yet determined. We then decompose all of the O(10, 10) potentials into type IIA/type IIB potentials. Then, since the type IIB potentials with level ℓ 9 ≤ 14 are already obtained in Table 2 with level ℓ 9 = 15 , whose multiplicity has not determined. Therefore, the multiplicity for the singlet is not determined in this paper.
There is an additional subtlety other than the O(10, 10) singlet. In that case, we could also determine the multiplicity of the O(10, 10) singlet H . In the following, we show the results, and there, the multiplicities of the mixed-symmetry potentials other than those colored in blue will be reliable.
In [37], the E 8 U -duality multiplets of exotic branes are searched by using the T -and Sduality, and we take a similar method here. 10 Starting with a standard potential, such as the R-R potential, we repeatedly perform T -duality (2.32) and S-duality (1.25) and obtain a chain of the underlined potentials, which we call SUSY potentials here. Of course, this procedure generates infinitely many SUSY potentials, and we need to cut off the potentials at a certain level. By using the appended mathematica notebook, we have checked that we can determine all of the SUSY potentials, at least up to level n = 36 . 11 Then, among the allowed O (10,10) potentials generated by SimpLie, we have identified which O(10, 10) potentials indeed appear with multiplicity 1, by comparing with a list of mixed-symmetry potentials.
We have determined all of the SUSY potentials up to n = 36 , but here we show the results only up to level n = 11 . The results up to level n = 24 (potential Z) are given in the ancillary files on arXiv. In addition, for potentials with level n ≥ 1 , we display only the type II mixed-symmetry potentials associated with the highest-weights and the number of the potentials (with different tensor structures) contained in the O(10, 10) multiplet is given in the square bracket. For example, at level n = 1, there is only one O(10, 10) multiplet Cȧ and it contains 5 type IIA potentials C 1 , C 3 , C 5 , C 7 , C 9 and 6 type IIB potentials C 0 , C 2 , C 4 , C 6 , C 8 , C 10 . The detailed contents for the higher levels can be easily generated by using the appended mathematica notebook.
In this paper, we determined all of the SUSY potentials up to level n = 6 , and they are precisely the same as those obtained here. Namely, at least for n ≤ 6 , all of the underlined E 11 potentials are in a single orbit of Weyl reflections (T -/S-dualities). We expect that this is the case also for the higher-level potentials, and the SUSY potentials obtained in this appendix will be all of the SUSY potentials (with n ≤ 24) predicted from the E 11 conjecture. 10 Here, we perform the T -duality even in the timelike direction, and under that T -duality, type II theory may be mapped to type II * theory but we do not distinguish type II and type II * carefully. At least, the results obtained here are totally consistent with the E11 conjecture. 11 A similar result up to level n = 25 has been obtained [80] but the reported number of potentials is much smaller than that obtained here, and their analysis may be restricted to a certain subclass. For example, as explained in footnote 6, L10,10,7,1 (which is predicted from E11) was not considered there.  [14] [0,1,0,1,0,0,0,0,0,0] [0,0,1,2,4,6,8,10,7,4,5] F 8,6 [9]