Exotic branes and mixed-symmetry potentials II: Duality rules and exceptional p -form gauge fields
aa r X i v : . [ h e p - t h ] A p r Exotic branes and mixed-symmetry potentials II:Duality rules and exceptional p -form gauge fields Jos´e J. Fern´andez-Melgarejo a ∗ , Yuho Sakatani b † , Shozo Uehara b ‡ a Departamento de F´ısica, Universidad de Murcia,Campus de Espinardo, 30100 Murcia, Spain b Department of Physics, Kyoto Prefectural University of Medicine,Kyoto 606-0823, Japan
Abstract In U -duality-manifest formulations, supergravity fields are packaged into covariant ob-jects such as the generalized metric and p -form fields A I p p . While a parameterization ofthe generalized metric in terms of supergravity fields is known for U -duality groups E n with n ≤
8, a parameterization of A I p p has not been fully determined. In this paper, wepropose a systematic method to determine the parameterization of A I p p , which necessarilyinvolves mixed-symmetry potentials. We also show how to systematically obtain the T -and S -duality transformation rules of the mixed-symmetry potentials entering the multi-plet. As the simplest non-trivial application, we find the parameterization and the dualityrules associated with the dual graviton. Additionally, we show that the 1-form field A I can be regarded as the generalized graviphoton in the exceptional spacetime. ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] ontents A I T -duality rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 S -duality rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 A Iµ as the generalized graviphoton . . . . . . . . . . . . . . . . . . . . . 18 A I p p
215 Summary and discussion 23A Notation 24B E n generators 24 B.1 M-theory parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25B.2 Type IIB parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Introduction
In the previous paper [1], we conducted a detailed survey of mixed-symmetry potentials in11D and type II supergravities. By considering their reduction to d dimensions, they yieldvarious p -form fields A I p p , which transform covariantly under E n U -duality transformation( n = 11 − d ). In the U -duality-covariant formulation of supergravity known as exceptionalfield theory (EFT) [2–5] (see [6–14] for earlier fundamental works), and the U -duality-manifestapproaches to brane actions [15–18], the p -form fields A I p p play an important role in providing U -duality-covariant descriptions. However, to make contact with the standard descriptions insupergravity and brane actions, explicit parameterizations of A I p p are needed. In this paper,we propose a systematic method to determine the parameterization of A I p p by utilizing theequivalence 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 ofvarious potentials can also be obtained. As the first non-trivial example, we obtain the T -and S -duality rules for the dual graviton, Eqs. (2.50)–(2.53) and (2.65), respectively.In EFT, the fundamental fields are the generalized metric M IJ and p -form fields A I p p , aswell as certain auxiliary fields. For E n EFT with n ≤ E n EFT with n ≤ M IJ , and as was concretely realized in [22], we can relate the two parameterizationsthrough some redefinitions of fields. As was shown in [22], by rewriting the M-theory fieldsin terms of type IIA fields, these field redefinitions are precisely the T -duality transformationrule. However, the analysis of [22] is limited to the E n EFT with n ≤ p -form potentials. In this paper, we extend their analysisto the case of E EFT, and find the T -duality and S -duality rules for the dual graviton.This gives a non-trivial check of our duality rules for the dual graviton mentioned in the firstparagraph.If we look at the explicit parameterization of the 1-form field A I , its first component A i isthe graviphoton. In 11D, the graviphoton is defined as ˆ A iµ ≡ ˆ g µν ˆ g νi , by using the 11D inversemetric ˆ g ˆ M ˆ N and the metric ˆ g µν in the external spacetime. In this paper, we propose that the1-form field A I can be regarded as a generalized graviphoton in the exceptional spacetime A Iµ = m µν M νI , m ≡ ( M µν ) − , (1.1)where M ˆ I ˆ J is the inverse generalized metric in E EFT (see [23] for a recent work) and ˆ I isthe index for the l -representation [6] of E that contains { µ, I } as a subset. We also findthat the parameterizations of the higher p -form fields A I p p ( p ≥
2) can be easily obtained fromthat of the 1-form A I through a simple antisymmetrization of indices.2 Parameterization of the 1-form A I In this section, we explain our method to determine the parameterizations of the 1-form A I .The index I transforms in a fundamental representation of the E n algebra with a Dynkin label[1 , , . . . , E n Dynkin diagram, M-theory and type IIB theory (see [1] and references therein for details): (/).*-+, α (/).*-+, α · · · M-theory (/).*-+, α n − (/).*-+, α n − × (/).*-+, α n (/).*-+, α n − (/).*-+, α n − , (/).*-+, α (/).*-+, α · · · Type IIB theory (/).*-+, α n − (/).*-+, α n − (/).*-+, α n × (/).*-+, α n − (/).*-+, α n − . (2.1)As is explained in the accompanying paper [1], in terms of M-theory, the 1-form field A I isdecomposed into SL( n ) tensors as follows: (cid:0) A Iµ (cid:1) = (cid:0) A iµ , A µ ; i i √ , A µ ; i ··· i √ , A µ ; i ··· i ,i √ , . . . (cid:1) , (2.2)where i, j = d, . . . , , z are indices of the fundamental representation of SL( n ). On the otherhand, in terms of type IIB theory, the 1-form field is decomposed into SL( n − × SL(2)tensors as follows: (cid:0) A I µ (cid:1) = (cid:0) A m µ , A αµ ; m , A µ ; m m m √ , A αµ ; m ··· m √ , A µ ; m ··· m , m √ , . . . (cid:1) , (2.3)where α, β = 1 , S -duality indices and m , n = d, . . . , n − A I µ . Althoughwe know the tensor structures of each component, it is not obvious how to determine theexplicit parameterization in terms of the standard supergravity fields, which is the main subjectof this paper.As demonstrated in [22], the two decompositions (2.2) and (2.3) can be related by usingthe equivalence between M-theory on T with coordinates ( x α ) = ( x y , x z ) and type IIB theoryon S with a coordinate x y :M-theory/ T x z (cid:15) (cid:15) l l our map , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ Type IIA theory/ S o o T -duality along x y / x y / / Type IIB theory/ S . (2.4)Here, x z is a coordinate along the M-theory circle, and the coordinate x y in M-theory (or typeIIA theory) is mapped to the coordinate x y in type IIB theory under the T -duality. By usingthe map, we can rewrite various quantities in M-theory in terms of type IIB supergravity. In order to discuss the parameterization, we will briefly explain the supergravity fields con-sidered in this paper. We basically follow the convention of [22].3
1D supergravity:
In 11D supergravity, we consider the following bosonic fields, { ˆ g ˆ M ˆ N , ˆ A ˆ3 , ˆ A ˆ6 , ˆ A ˆ8 , ˆ1 } (cid:0) ˆ M , ˆ N = 0 , . . . , , z (cid:1) . (2.5)The standard potentials ˆ A and ˆ A couple to the M2-brane and M5-brane, respectively, whilethe dual graviton ˆ A , couples to the Kaluza–Klein monopole 6 (sometimes called MKK) [24].When we consider a compactification to d dimensions, the 11D metric ˆ g ˆ M ˆ N is decomposed as(ˆ g ˆ M ˆ N ) = ˆ g µν + ˆ A kµ ˆ G kl ˆ A lν − ˆ A kµ ˆ G kj − ˆ G ik ˆ A kν ˆ G ij ( µ, ν = 0 , . . . , d − , (2.6)where we have defined the graviphoton as ˆ A iµ ≡ − ˆ g µk ˆ G ki = ˆ g µν ˆ g νi . Type IIA supergravity:
When we consider type IIA supergravity, we use the followingstandard 11D–10D map:(ˆ g ˆ M ˆ N ) ≡ ˆ g MN ˆ g Mz ˆ g zN ˆ g zz = e − Φ g MN + e Φ C M C N e Φ C M e Φ C N e Φ , ˆ A ˆ3 = C + B ∧ d x z , ˆ A ˆ6 = B + (cid:0) C − C ∧ B (cid:1) ∧ d x z , (2.7)where we have added the hat to the subscript, like ˆ A ˆ p , to stress that it is a p -form in 11D. Inour convention, the dual graviton ˆ A ˆ8 , ˆ1 = { ˆ A ˆ8 , , ˆ A ˆ8 ,z } follows the 11D–10D map,ˆ A ˆ8 , = A , + A , ∧ d x z , ˆ A ˆ8 ,z = A + (cid:0) C − C ∧ B ∧ B (cid:1) ∧ d x z , (2.8)where ˆ A ˆ8 ,z corresponds to ˆ˜ N studied in [24]. The metric and the graviphoton are defined as( g MN ) = g µν + A pµ G pq A qν − A pµ G pn − G mp A pν G mn , A mµ ≡ g µν g νm . (2.9)Then we find the 11D–10D map for the graviphoton:ˆ A mµ = A mµ , ˆ A zµ = − (cid:0) C µ + A pµ C p (cid:1) . (2.10) Type IIB supergravity:
In type IIB theory, in addition to the standard Einstein-framemetric g MN , we consider the following SL(2) S -duality-covariant tensors:( m αβ ) ≡ e ϕ e − ϕ +( C ) C C , ( A α ) ≡ B − C , (2.11) A ≡ C − C ∧ B , ( A α ) ≡ C − C ∧ B + B ∧ C ∧ B − (cid:0) B − C ∧ C + B ∧ C ∧ C (cid:1) . (2.12)We also consider the dual graviton A , , whose behavior under duality transformations is tobe determined. Upon compactification to d dimensions, the graviphoton is introduced as( g MN ) = g µν + A p µ G pq A q ν − A p µ G pn − G mp A p ν G mn , A m µ ≡ g µν g ν m . (2.13)4 .2 Strategy: Linear map Here, let us explain the detailed procedure, how to determine the parameterization of the1-form in both the M-theory and type IIB languages:( A Iµ ) = A iµ A µ ; i i √ A µ ; i ··· i √ A µ ; i ··· i ,i √ ... , ( A I µ ) = A m µ A αµ ; m A µ ; m m m √ A αµ ; m ··· m √ A µ ; m ··· m , m √ ... , (2.14)where ellipses stand for the rest of the components that complete the U -duality multiplet thatpotentially involve further mixed-symmetry potentials.To determine the parameterization, we make the following modest assumptions: • The M-theory fields A p,q,r,... and the type IIB fields A α ··· α s p,q,r,... are respectively param-eterized by the following fields:M-theory: { ˆ A iµ , ˆ A ˆ3 , ˆ A ˆ6 , ˆ A ˆ8 , ˆ1 , . . . } , (2.15)Type IIB theory: { A m µ , A α , A , A α , A , , . . . } . (2.16) • The top form is normalized with weight one: A µ ; p,q,r,... = ˆ A µp,q,r,... + (sum of products of potentials) , A α ··· α s µ ; p,q,r,... = A α ··· α s µp,q,r,... + (sum of products of potentials) . (2.17)According to these, the first components of the 1-forms should be, respectively, A iµ = ˆ A iµ (M-theory) , A m µ = A m µ (type IIB) . (2.18)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 coordinateson the n -torus in M-theory as ( x i ) = ( x a , x α ) ( a, b = 1 , . . . , n −
2) and those on the ( n − x m ) = ( x a , x y ) . Under the decomposition, the 1-form fields (2.14)are decomposed into SL( n − × SL(2) tensors as follows:( A Iµ ) = A aµ A αµ A µ ; a a √ A µ ; aα A µ ; yz A µ ; a ··· a √ A µ ; a ··· a α √ A µ ; a a a yz √ A µ ; a ··· a yz,a √ A µ ; a ··· a yz,α √ ... , ( A I µ ) = A aµ A y µ A αµ ; a A αµ ; y A µ ; a a a √ A µ ; a a y √ A αµ ; a ··· a √ A αµ ; a ··· a y √ A µ ; a ··· a y ,a √ A µ ; a ··· a y , y √ ... , (2.19)5here toroidal directions (either compactified or T -dualized) are shown explicitly. In terms ofthe Dynkin diagram given in (2.1), in M-theory we have first performed the level decompositionassociated with the node α n . Secondly, we have done the level decomposition associated with α n − . On the other hand, in type IIB theory the order is reversed. In the end, we obtain thesame decomposition. Indeed, the set of SL( n − × SL(2) tensors appearing in (2.19) has thesame structure. Then, we make the following identifications [22]: A aµ A αµ A µa a √ A µ ; aα A µ ; yz A µ ; a ··· a √ A µ ; a ··· a α √ A µ ; a a a yz √ A µ ; a ··· a yz,a √ A µ ; a ··· a yz,α √ ... M = A aµ A αµ ; y A µ ; a a y √ A βµ ; a ǫ βα A y µ A µ ; a ··· a y , y √ A βµ ; a ··· a ǫ βα √ A µ ; a a a √ A µ ; a ··· a y ,a √ A βµ ; a ··· a ǫ βα √ ... IIB or A aµ A y µ A αµ ; a A αµ ; y A µ ; a a a √ A µ ; a a y √ A αµ ; a ··· a √ A αµ ; a ··· a y √ A µ ; a ··· a y ,a √ A µ ; a ··· a y , y √ ... IIB = A aµ A µ ; yz ǫ αβ A µ ; aβ A αµ A µ ; a a a yz √ A µ ; a a √ ǫ αβ A µ ; a ··· a yz,β √ ǫ αβ A µ ; a ··· a β √ A µ ; a ··· a yz,a √ A µ ; a ··· a √ ... M , (2.20)where we have defined ǫ ≡ ( ǫ αβ ) ≡ ( ǫ αβ ) ≡ − . (2.21)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 A Iµ = S I J A J µ , A I µ = ( S − ) I J A Jµ . (2.22)We note that this identification was originally proposed in [25] in the context of E .Now, for simplicity, we assume the standard T -duality rule for the NS–NS fields g AB A–B = g AB − g A y g B y − B A y B B y g yy , g Ay A–B = − B A y g yy , g yy A–B = 1 g yy , B AB A–B = B AB − B A y g B y − g A y B B y g yy , B Ay A–B = − g A y g yy , (2.23)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: A aµ A–B = A aµ , A yµ A–B = B µ y + A p µ B py . (2.24)By using the 11D–10D relation (2.10), the first rule gives A aµ = ˆ A aµ M–A = A aµ A–B = A aµ = A aµ , (2.25)which is nothing but the first row of (2.20). This assumption is not necessary in the approach discussed in Section 3. .3 Detailed procedures We will continue this process by considering the index structure. The second components ofthe 1-form in the M-theory and the type IIB parameterization are generically expanded as A µ ; i i = ˆ A µi i + c ˆ A kµ ˆ A ki i , A αµ ; m = A αµ m + c A p µ A α pm , (2.26)where c and c are parameters to be determined. From A αµ M–B = A αµ ; y in (2.20), we haveˆ A αµ = A αµ M–B = A αµ ; y = A αµ y + c A p µ A α py . (2.27)On the other hand, the second rule of (2.24) and the 11D–10D relation (2.10) givesˆ A yµ M–B = A yµ A–B = B µ y + A p µ B py , (2.28)and by comparing this with the α = y component of (2.27) , we find c = 1 .Similarly, the map A µ ; yz M–B = A y µ in (2.20) gives A µ ; yz = ˆ A µyz + c ˆ A aµ ˆ A ayz M–B = A y µ . (2.29)On the other hand, the second line of (2.23) and the 11D–10D relation (2.7) giveˆ A ABz
M–B = B AB − B A y g B y − g A y B B y g yy , ˆ A Ayz
M–B = − g A y g yy . (2.30)By substituting the second relation into the left-hand side of (2.29) and using g µ y = − ( A aµ g a y + A y µ g yy ) , we obtain A aµ g a y + A y µ g yy − c A aµ g a y g yy B–M = ˆ A µyz + c ˆ A aµ ˆ A ayz M–B = A y µ = A y µ , (2.31)which shows c = 1 . Thus, the parameterizations of A µ ; i i and A αµ ; m are determined as A µ ; ij = ˆ A µij + ˆ A kµ ˆ A kij , A αµ ; m = A αµ m + A p µ A α pm . (2.32)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 A y g yy and thegraviphoton does not appear explicitly.By using these assumptions, we obtain the set of standard T -duality rules.7or example, from the α = z component of (2.27), we findˆ A zµ M–B = − C µ y − A p µ C py . (2.33)In terms of the type IIA field, this is equivalent to C µ + A aµ C a + A yµ C y A–B = C µ y + A aµ C a y , (2.34)and by using the identity g µy = − ( A aµ g ay + A yµ g yy ) , we obtain (cid:16) C µ − g µy g yy C y (cid:17) + A aµ (cid:16) C a − g ay g yy C y (cid:17) A–B = C µ y + A aµ C a y . (2.35)From the assumption that the T -duality rule does not contain the graviphoton explicitly, thisimplies the standard T -duality rule: C A y B-A = C A − C y g Ay g yy , (2.36)or conversely, C A A–B = C A y − C B A y , (2.37)where we have employed the standard rule C y A–B = C .Similarly, if we consider the linear map A µ ; aα M–B = A βµ ; a ǫ βα in (2.20), we findˆ A µaα + ˆ A kµ ˆ A kaα M–B = (cid:0) A βµa + A p µ A β p a (cid:1) ǫ βα . (2.38)In particular, for α = y , we obtain a map between the type IIA/IIB fields, C µay + A bµ C bay − (cid:0) C µ + A mµ C m (cid:1) B ay A–B = C µa + A bµ C ba + A y µ C y a , (2.39)and this is equivalent to C µay − C µ B ay + C y B ay g µy g yy + A bµ (cid:16) C bay − C b B ay + C y B ay g by g yy (cid:17) A–B = C µa − C y a g µ y g yy + A bµ (cid:16) C ba − C y a g b y g yy (cid:17) . (2.40)Then, we find the T -duality rule C ABy
A–B = C AB − C [ A | y g | B ] y g yy . (2.41) Further steps
We can further proceed by considering a general expansion of the SL(2) singlet A µ ; m m m , A µ ; m m m = A µ m m m + c ǫ αβ A αµ [ m A β m m ] + c A p µ A pm m m + c ǫ αβ A p µ A α p [ m A β m m ] . (2.42)Similarly, unknown T -duality rules can also be expanded by considering possible 9D covariantexpressions with parameters. Then, the consistency with the linear map (2.20) determinesall of the parameters. In this manner, by using the linear map (2.20), we can find both theparameterization of A Iµ and T -duality rules for the gauge potentials one after another.8 .4 Results By continuing the above procedure, we have determined the M-theory parameterization as( A Iµ ) = A iµ A µ ; i i √ A µ ; i ··· i √ A µ ; i ··· i ,i √ ... = ˆ A iµ √ (cid:0) ˆ N µ ; i i + ˆ A kµ ˆ N k ; i i (cid:1) √ (cid:0) ˆ N µ ; i ··· i + ˆ A kµ ˆ N k ; i ··· i (cid:1) √ (cid:0) ˆ N µ ; i ··· i ,i + ˆ A kµ ˆ N k ; i ··· i ,i (cid:1) ... . (2.43)Remarkably, the two tensors ˆ N µ ; p,q,r,... and ˆ N k ; p,q,r,... in each row can be regarded as particularcomponents of 11D-covariant tensors:ˆ N ˆ M ; ˆ M ˆ M = ˆ A ˆ M ˆ M ˆ M , ˆ N ˆ M ; ˆ M ··· ˆ M = ˆ A ˆ M ··· ˆ M − A ˆ M [ ˆ M ˆ M ˆ A ˆ M ˆ M ˆ M ] , ˆ N ˆ M ; ˆ M ··· ˆ M , ˆ N ≃ ˆ A ˆ M ··· ˆ M , ˆ N − (cid:0) ˆ A ˆ M [ ˆ M ··· ˆ M ˆ A ˆ M ˆ M ] ˆ N − ˆ A ˆ M [ ˆ M ··· ˆ M ˆ A ˆ M ˆ M ˆ N ] (cid:1) + 35 ˆ A ˆ M [ ˆ M ˆ M ˆ A ˆ M ˆ M ˆ M ˆ A ˆ M ˆ M ] ˆ N , (2.44)where the meaning of the equivalence ≃ is explained below.As discussed in [26–29] (see also [1]), for any mixed-symmetry potential, not all of thecomponents couple to supersymmetric branes. For the dual graviton A µ ; i ··· i ,i , only thecomponents satisfying i ∈ { i , . . . , i } (2.45)couple to supersymmetric branes. The components that do not couple to supersymmetricbranes correspond to the E roots α satisfying α · α < p -form potentials under T -duality and S -duality. Since our procedure to determinethe parameterization is based on T -duality and S -duality, it can only provide the parame-terization of the components that couple to supersymmetric branes. In this sense, it is morehonest to express the last equation of (2.44) asˆ N ˆ M ; ˆ M ··· ˆ M x,x = ˆ A ˆ M ··· ˆ M x,x −
21 ˆ A ˆ M [ ˆ M ··· ˆ M ˆ A ˆ M x ] x + 35 ˆ A ˆ M [ ˆ M ˆ M ˆ A ˆ M ˆ M ˆ M ˆ A ˆ M x ] x = ˆ A ˆ M ··· ˆ M x,x −
15 ˆ A ˆ M x [ ˆ M ··· ˆ M ˆ A ˆ M ˆ M ] x −
10 ˆ A ˆ M x [ ˆ M ˆ A ˆ M ˆ M ˆ M ˆ A ˆ M ˆ M ] x + 15 ˆ A ˆ M [ ˆ M ˆ M ˆ A ˆ M ˆ M | x | ˆ A ˆ M ˆ M ] x . (2.46)In this paper, equalities that hold under the restriction (2.45) are denoted by ≃ . The param-eterizations of mixed-symmetry potentials that do not satisfy the restriction (2.45) are notdetermined in this paper. 9ow we turn to the results in type IIB theory. The parameterization takes the form( A I µ ) = A m µ (cid:0) N αµ ; m + A p µ N α p ; m (cid:1) √ (cid:0) N µ ; m m m + A p µ N p ; m m m (cid:1) √ (cid:0) N αµ ; m ··· m + A p µ N α p ; m ··· m (cid:1) √ (cid:0) N µ ; m ··· m , m + A p µ N p ; m ··· m , m (cid:1) ... , (2.47)where N αM ; M = A αM M , N M ; M M M ≡ A M ··· M − ǫ γδ A γM [ M A δM M ] = C M ··· M − C M [ M B M M ] , N αM ; M ··· M ≡ A αM ··· M + 5 A αM [ M A M ··· M ] + 5 ǫ γδ A γM [ M A δM M A αM M ] = C M ··· M − C M [ M M M B M M ] + 15 C M [ M B M M B M M ] − (cid:0) B M ··· M − C M [ M M M C M M ] (cid:1) , N M ; M ··· M ,N ≃ A M ··· M ,N + 6 B M [ M ··· M B M ] N − C M [ M C M ··· M ] N − C M [ M M M C M M B M ] N + 10 C M [ M M M C M M M ] N + 452 (cid:0) B M [ M B M M C M M C M ] N − C M [ M C M M B M M B M ] N (cid:1) . (2.48)The last component N , is relatively long, and the S -duality invariance is not clear. However,this is because of the definition of the dual graviton A , . As we will see later (in Eq. (3.47)),a certain redefinition of A , makes the expression of N , simpler. T -duality rule In addition to the parameterizations, we have obtained the T -duality rules as follows: g AB A–B = g AB − g A y g B y − B A y B B y g yy , g Ay A–B = − B A y g yy , A aµ A–B = A aµ , A yµ A–B = B µ y + A p µ B py , B AB A–B = B AB − B A y g B y − g A y B B y g yy , B Ay A–B = − g A y g yy , C A ··· A n − y A–B = C A ··· A n − − ( n − C [ A ··· A n − | y | g A n − ] y g yy , C A ··· A n A–B = C A ··· A n y − n C [ A ··· A n − B A n ] y − n ( n − C [ A ··· A n − | y | B A n − | y | g A n ] y g yy . (2.49)10or the 6-form potential B and the dual graviton A , , we find B A ··· A y A–B = B A ··· A y − A [ A ··· A C A ] y − A [ A A A | y | C A A ] − C [ A A B A A C A ] y − C [ A A C A A B A ] y − A [ A ··· A | y | C A | y | g A ] y g yy − C [ A A B A | y | C A | y | g A ] y g yy , (2.50) B A ··· A A–B = A A ··· A y , y − B [ A ··· A | y | B A ] y + 30 A [ A A A | y C | A A B A ] y + 30 A [ A ··· A C A | y | B A ] y − B [ A A B A | y | C A A C A ] y + 60 A [ A A A | y B A | y C A | y g | A ] y g yy , (2.51) A A ··· A y,y A–B = B A ··· A − B [ A A C A A C A A ] − B [ A ··· A | y g | A ] y g yy + 20 A [ A A A | y | C A A g A ] y g yy + 30 B [ A | y | C A A C A A g A ] y g yy + 150 B [ A A C A A C A | y | g A ] y g yy , (2.52) A A ··· A y,B A–B ≃ A A ··· A y ,B − C [ A ··· A | y | C A ] B − B [ A ··· A | B | B A ] y + 10 C [ A A A | B | C A A A ] y + 20 C [ A A A | y | B A | B | C A A ] + 40 C [ A A A | y | B A A C A ] B + 30 B [ A A C A A C A A ] B B y + 452 B [ A A C A A (cid:0) − B A | y | C A ] B + C A | y | B A ] B (cid:1) − A A ··· A y , y g B y g yy + 6 C [ A ··· A | y | C A ] y g B y g yy − C B y C [ A ··· A | y | g | A ] y g yy + 6 B B y B [ A ··· A | y g | A ] y g yy − C [ A A | B y C | A A A | y | g A ] y g yy + 120 C [ A A | B y | B A A C A | y | g A ] y g yy − B B y C [ A A A | y | C A A g A ] y g yy + 40 C [ A A A | y | B A | B | C A | y | g A ] y g yy − B B y B [ A A C A A C A | y g | A ] y g yy + 452 B [ A A B A | y | C A A C A ] y g B y g yy . (2.53)The T -duality rules (2.50) and (2.52) coincide with the known results [30] (see Appendix Atherein), for which the following identification of supergravity fields is needed: g µν BC (1) C (3) C (5) ˜ BN (IIA)[30] = g MN B C C C − B A , (IIA)here , g µν B C (0) C (2) C (4) C (6) ˜ B (IIB)[30] = g MN B − C − C − A − (cid:0) C − B ∧ B ∧ C (cid:1) − (cid:0) B − C ∧ C ∧ B (cid:1) (IIB)here . (2.54)11n the other hand, (2.53) has been obtained in [31], where B = 0 and C = 0 are assumed.If we truncate B and C , we have A = C and the T -duality rule (2.53) reduces to A A ··· A y,B A–B ≃ A A ··· A y ,B + 10 A [ A A A | B | A A A A ] y − A A ··· A y , y g B y g yy − A [ A A | B y A | A A A | y | g A ] y g yy . (2.55)More explicitly, according to the restriction (2.45), the direction x ≡ B must be contained in { A , . . . , A } and by choosing A = x , we have A A ··· A xy,x A–B = A A ··· A x y ,x + 5 A [ A A A | x | A A A ] x y − A A ··· A x y , y g x y g yy + 5 A [ A A | x y A | A A | x y | g A ] y g yy − A [ A A | x y A | A A A ] y g x y g yy . (2.56)If we denote k ≡ ∂ x and h ≡ ∂ y , and define N (7) M ··· M ≡ A M ··· M ,x , N (7) M ··· M ≡ A M ··· M ,x , N (7) M ··· M ≡ A M ··· M , y , (2.57)the result of [31] (see Eq. (5.13)) is precisely reproduced:( ι k N (7) ) A ··· A y A–B = ( ι k ι h N (7) ) A ··· A − ι k A ) [ A A A ( ι k ι h A ) A A ] − ( ι k ι h N (7) ) A ··· A g x y g yy − ι k ι h A ) [ A A ( ι k ι h A ) A A g A ] y g yy + 5 ( ι h A ) [ A A A ( ι k ι h A ) A A ] g x y g yy . (2.58)This shows that A , corresponds to the 7-form N (7) or N (7) of [31] under B = 0 and C = 0.In the above computation, we have shown only T -duality transformations from type IIBto type IIA, but we can easily find the inverse map. The standard rules (2.49) have the sameform even for the map from type IIA to type IIB: g AB B–A = g AB − g Ay g By − B Ay B By g yy , g A y B–A = − B Ay g yy , A aµ B–A = A aµ , A y µ B–A = B µy + A pµ B py , B AB B–A = B AB − B Ay g By − g Ay B By g yy , B Ay B–A = − g Ay g yy , C A ··· A n − y B–A = C A ··· A n − − ( n − C [ A ··· A n − | y | g A n − ] y g yy , (2.59) C A ··· A n B–A = C A ··· A n y − n C [ A ··· A n − B A n ] y − n ( n − C [ A ··· A n − | y | B A n − | y | g A n ] y g yy . Regarding the 6-form potential and the dual graviton, the results are as follows: B A ··· A y B–A = B A ··· A y + 5 C [ A ··· A | y | C A ] + 5 C [ A A A C A A ] y − C [ A A | y | C A A | y | g A ] y g yy − C y C [ A ··· A | y | g A ] y g yy , (2.60) B A ··· A B–A = A A ··· A y,y − B [ A ··· A y B A ] y − C [ A ··· A | y | C A B A ] y C [ A A A C A A | y | B A ] y + 30 C [ A A | y | C A A | y | B A A ] − C y C [ A ··· A | y | B A | y | g A ] y g yy − C [ A A | y | C A A | y | B A | y | g A ] y g yy , (2.61) A A ··· A y , y B–A = B A ··· A − B [ A ··· A | y | g A ] y g yy + 452 C [ A A | y | B A A C A g A ] y g yy , (2.62) A A ··· A y ,B B–A ≃ A A ··· A y,B − B [ A ··· A | B | B A ] y + 6 C [ A ··· A C A ] By − C [ A ··· A | y | C A A ] B + 20 C [ A A A C A A | B | B A ] y − C [ A A A C A A | y | B A ] B − C [ A A A C A | By | B A A ] − C [ A A | y | C A B A | B | B A A ] − A [ A ··· A | By,y g | A ] y g yy (2.63)+ 6 B By B [ A ··· A | y | g A ] y g yy + 15 C [ A ··· A | y | C A | By | g A ] y g yy + 152 C [ A ··· A | y | C A A ] y g By g yy − C [ A A A C A A | y | B A ] y g By g yy − C [ A A | y | C A A | y | B A A ] g By g yy − C [ A A | y | C A A | y | B A | B | g A ] y g yy − B By C [ A A | y | C A B A A g A ] y g yy − C y C [ A A | y | B A A B A | B | g A ] y g yy . Now, let us comment more on the restriction rule. In the T -duality rule (2.53), we areassuming that B is contained in { A , · · · , A } . When the restriction is removed, we expectthat the right-hand side of the T -duality rule is modified. In general, the components thatdo not satisfy the restriction are in the same orbit as the ( α = β )-component of the type IIBpotential A αβ , which is electric-magnetic dual to the 0-form potential m αβ . Therefore, it willbe possible that A αβA ··· A B y appears on the right-hand side of (2.53). S -duality rule The standard S -duality transformation rules are reproduced as follows: g ′ MN = g MN , A ′ m µ = A m µ , C ′ = − C ( C ) + e − ϕ , e − ϕ ′ = e − ϕ ′ ( C ) + e − ϕ , B ′ = − C , C ′ = B , C ′ = C − B ∧ C , A ′ = A , C ′ = − B + 12 B ∧ C ∧ C , B ′ = C − C ∧ B ∧ B , A ′ = A , A ′ = − A . (2.64)From the S -duality invariance of A µ ; m ··· m , m , we also find A ′ M ··· M ,M ≃ A M ··· M ,M + 7 (cid:0) B [ M ··· M B M ] M − B [ M ··· M B M M ] (cid:1) + 7 (cid:0) C [ M ··· M C M ] M − C [ M ··· M C M M ] (cid:1) − (cid:0) A [ M ··· M B M M C M ] M − A [ M ··· M C M M B M M ] (cid:1) + 9454 (cid:0) B [ M M B M M C M M C M ] M − B [ M M B M M C M M C M M ] (cid:1) . (2.65)13 Another approach based on the generalized metric
In this section, we discuss another derivation of the T -/ S -duality transformation rule for thedual graviton, which is based on the generalized metric. We also explain another method todetermine the parameterization of the 1-form A Iµ .In d dimensions, scalar fields are packaged into a U -duality-covariant object called thegeneralized metric, denoted as M IJ and M IJ in M-theory and type IIB, respectively. Thegeneralized vielbeins, E I J and E IJ respectively, are defined such that M IJ ≡ δ KL E K I E LJ , M IJ ≡ δ KL E KI E LJ . (3.1)According to [13], the generalized vielbein can be constructed as follows. We first considerthe positive-root generators of the E n algebra, which are summarized as { E α } = { K ij ( i < j ) , R i i i , R i ··· i , R i ··· i ,i , . . . (cid:9) (3.2)in the M-theory parameterization and as { E α } = { K mn ( m < n ) , R , R m m α , R m ··· m , R m ··· m α , R m ··· m , m , . . . } (3.3)in the type IIB parameterization. We also consider the Cartan generators, { H k } = { K dd − K d +1 d +1 , . . . , K − K zz , K + K + K zz + D } (3.4)in the M-theory parameterization ( D ≡ K ii ) and { H k } = { K dd − K d +1 d +1 , . . . , K − K , K + K − D − R , R , K − K } (3.5)in the type IIB parameterization ( D ≡ K mm ). Then, we prepare the matrix representations ofthese generators in the vector representation. In the M-theory parameterization, the matrixrepresentations have been obtained in [13] for n ≤ n = 8 . In the type IIB pa-rameterization, they have been determined in [20,21] for n ≤
7. The results for n = 8 are givenin Appendix B. Then, we define the generalized vielbein in the M-theory parameterization as E ≡ ( E I J ) ≡ ˆ E L , ˆ E ≡ e h k H k e P i 00 0 0 ˆ G i ··· i ,j ··· j ˆ G ij ... . .. , (3.10)ˆ M ≡ | G | d − G mn m αβ G mn G m m m , n n n · · · m αβ G m ··· m , n ··· n 00 0 0 0 G m ··· m , n ··· n G mn ... . . . , (3.11)where ˆ G i ··· i p ,j ··· j p ≡ δ j ··· j p k ··· k p ˆ G i k · · · ˆ G i p k p , | ˆ G | ≡ det( ˆ G ij ) , G m ··· m p , n ··· n p ≡ δ n ··· n p q ··· q p G m q · · · G m p q p , | G | ≡ det( G mn ) . (3.12)On the other hand, the twist matrices L and L contain various gauge potentials, which can becomputed 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 theprocedure to obtain the duality rules, which has been proposed in [22] for n ≤ Here, we explain how to determine the duality transformation rules from the generalizedmetric. As we have discussed in Section 2, in the M-theory and type IIB parameterizations,we are using different bases, which are related through the linear map (2.22). Accordingly,the generalized metrics in the two parameterizations are related as M IJ = ( S ⊺ ) I K M KL S L J . (3.13)The explicit form of S I J has been obtained in [22] only for n ≤ n ≤ x I = S I J x J , x I = ( S − ) I J x J . (3.14)15ince the matrix size of S I J is very large (21 × x a x αy a a √ y aα y yzy a ··· a √ y a ··· a α √ y a a a yz √ y a ··· a α,a √ y a ··· a α,β ) √ ǫ αβ y a ··· a α,β ] √ 2! 6! y a ··· a yz,a √ y a ··· a yz,α √ y a ··· a yz,b b b √ 6! 3! y a ··· a yz,b b α √ 6! 2! y a ··· a yz,ayz √ y a ··· a yz,b ··· b √ 6! 6! y a ··· a yz,b ··· b α √ 6! 5! y a ··· a yz,b ··· b yz √ 6! 4! y a ··· a yz,b ··· b yz,a √ 6! 6! y a ··· a yz,b ··· b yz,α √ 6! 6! | {z } x I = x a y α yy a a y √ y βa ǫ βα x yy a ··· a y , y √ y βa ··· a ǫ βα √ y a a a √ y βa ··· a y ,a y ǫ βα √ ǫ αγ ǫ βδ y γδa ··· a y √ Y a ··· a Y a ··· a ,a y βa ··· a ǫ βα √ y a ··· a y ,b b b y √ 6! 3! y γa ··· a y ,b b ǫ βα √ 6! 2! y a ··· a ,a √ y a ··· a y ,b ··· b y , y √ 6! 6! y βa ··· a y ,b ··· b y ǫ βα √ 6! 5! y a ··· a y ,b ··· b √ 6! 4! y a ··· a y ,b ··· b y ,a √ 6! 6! y βa ··· a y ,b ··· b ǫ βα √ 6! 6! | {z } S I I x I ⇔ x a x y y αa y α yy a a a √ y a a y √ y αa ··· a √ y αa ··· a y √ y a ··· a ,a √ y a ··· a , y √ y a ··· a y ,a √ y a ··· a y , y √ y αβa ··· a y √ y αa ··· a y ,b b √ 6! 2! y αa ··· a y ,a y √ y a ··· a y ,b ··· b √ 6! 4! y a ··· a y ,b b b y √ 6! 3! y αa ··· a y ,b ··· b √ 6! 6! y αa ··· a y ,b ··· b y √ 6! 5! y a ··· a y ,b ··· b y ,a √ 6! 6! y a ··· a y ,b ··· b y , y √ 6! 6! | {z } x I = x a y yz ǫ αβ y aβ x αy a a a yz √ y a a √ ǫ αβ y a ··· a yz,β √ ǫ αβ y a ··· a β √ y a ··· a yz,ayz √ Y a ··· a Y a ··· a ,ay a ··· a √ ǫ αγ ǫ βδ y a ··· a γ,δ ) √ ǫ αβ y a ··· a yz,b b β √ 6! 2! ǫ αβ y a ··· a β,a √ y a ··· a yz,b ··· b yz √ 6! 4! y a ··· a yz,b b b √ 6! 3! ǫ αβ y a ··· a yz,b ··· b yz,β √ 6! 6! ǫ αβ y a ··· a yz,b ··· b β √ 6! 5! y a ··· a yz,b ··· b yz,a √ 6! 6! y a ··· a yz,b ··· b √ 6! 6! | {z } ( S − ) I I x I , where Y a ··· a Y a ··· a ,a ≡ √ δ b ··· b a ··· a √ 6! 6! 3 √ − δ b ··· b ba ··· a √ 6! 5!3 √ − δ b ··· b a ··· a a √ 5! 6! δ b ··· b a ··· a δ ba − √ δ b ··· b ba ··· a a √ 5! 5! y b ··· b , y √ y b ··· b y ,b √ , (3.15) Y a ··· a Y a ··· a ,a ≡ √ δ b ··· b a ··· a √ 6! 6! 3 √ − δ b ··· b ba ··· a √ 6! 5!3 √ − δ b ··· b a ··· a a √ 5! 6! δ b ··· b a ··· a δ ba − √ δ b ··· b ba ··· a a √ 5! 5! ǫ αβ y b ··· b α,β ] √ 2! 6! y b ··· b yz,b √ (3.16)with δ j ··· j n i ··· i n ≡ n ! δ j ··· j n i ··· i n . The constant matrix S I J can be read off from the above map betweenthe coordinates. We can check that the matrix S I I satisfies the property S I K ( S T ) K J = δ IJ , ( S T ) I K S K J = δ IJ , (3.17)under the generalized transpose, which is defined for a matrix A = ( A I J ) as( A T ) I J ≡ δ IK ( A ⊺ ) K L δ LJ ≡ δ IK A LK δ LJ , (3.18)namely the standard matrix transpose ⊺ followed by a flip in the position of the indices. Thisproperty shows that the flat metric is preserved under the linear map: δ IJ = S K I S L J δ KL . (3.19)16ow, 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-theoryfields in terms of the type IIB fields, and vice versa. In the case n ≤ n ≥ E EFT. By comparing the two parameterizations (3.13) of the E generalized metric, we find thefollowing relation between the M-theory fields and type IIB fields: (cid:0) ˆ G ij (cid:1) = ˆ G ab ˆ G aβ ˆ G αb ˆ G αβ M–B = e − ϕ G / yy δ ca − A γa y δ γα G c y ,d y G yy e ϕ G yy m γδ δ db − A δb y δ δβ , (3.20) ˆ A ayz M–B = G a y G yy , (3.21) ˆ A abα M–B = (cid:16) A βab − A β [ a | y | G b ] y G yy (cid:17) ǫ βα , (3.22) ˆ A abc M–B = A abc y − ǫ γδ A γ [ ab A δc ] y − ǫ γδ A γ [ a | y | A δb | y | G c ] y G yy , (3.23) ˆ A a ··· a yz M–B = A a ··· a − (cid:16) A [ a a a | y | G a ] y G yy + 3 ǫ γδ A γ [ a a A δa | y | G a ] y G yy (cid:17) , (3.24) ˆ A a ··· a α M–B = (cid:16) A βa ··· a y + 5 A [ a a a | y | A βa a ] + 52 ǫ γδ A γ [ a a A δa | y | A βa a ] − A [ a a a | y | A βa | y | G a ] y G yy − ǫ γδ A γ [ a a A δa | y | A βa | y | G a ] y G yy (cid:17) ǫ βα (3.25) ˆ A a ··· a M–B = A a ··· a y , y − ǫ γδ A [ a a a | y | (cid:16) A γa a A δa ] y + 2 A γa | y | A δa | y | G a ] y G yy (cid:17) , (3.26) ˆ A a ··· a yz,α M–B = (cid:20) A βa ··· a + A β [ a a (cid:0) A a a a | y | + 30 ǫ γδ A γa a A δa | y | (cid:1) G a ] y G yy (cid:21) ǫ βα , (3.27) ˆ A a ··· a yz,b M–B = A a ··· a y ,b − A [ a ··· a A a a ] b y − ǫ αβ A [ a a a | y | A αa a A βa ] b + 152 ǫ αβ ǫ γδ A α [ a a A βa | b | A γa a A δa ] y + 20 A [ a a a | y | A a a | b y | G a ] y G yy − ǫ αβ A [ a a a | y | A αa | b | A βa | y | G a ] y G yy − ǫ αβ A [ a a a | y | A αa a A β | b y | G a ] y G yy + 30 ǫ αβ A [ a a | b y | A αa a A βa | y | G a ] y G yy + 15 ǫ αβ ǫ γδ (cid:0) A α [ a a A βa | b | A γa | y | A δa | y | − A α [ a a A βa | y | A γa a A δ | b y | (cid:1) G a ] y G yy . (3.28)17y making identifications ˆ A ˆ M ˆ M ˆ M = ˆ A ˆ M ˆ M ˆ M , ˆ A ˆ M ··· ˆ M = ˆ A ˆ M ··· ˆ M , ˆ A ˆ M ··· ˆ M , ˆ N ≃ ˆ A ˆ M ··· ˆ M , ˆ N − 28 ˆ A [ ˆ M ··· ˆ M ˆ A ˆ M ˆ M ] ˆ N (3.29)for M-theory fields and A αM M = A αM M , A M ··· M = A M ··· M , A αM ··· M = A αM ··· M , (3.30) A M ··· M ,N ≃ A M ··· M ,N + 7 (cid:0) B [ M ··· M B M ] N − B [ M ··· M B M N ] (cid:1) + 1054 (cid:2) C [ M ··· M (cid:0) B M M C M ] N − C M M B M ] N (cid:1) − C [ M ··· M (cid:0) B M M C M N ] − C M M B M N ] (cid:1)(cid:3) + 3158 (cid:0) C [ M M C M M B M M B M ] N − C [ M M C M M B M M B M N ] (cid:1) (3.31)for type IIB fields, and by using the 11D–10D map, these relations are precisely the T -dualityrules obtained in Section 2.5. The S -duality rule for the new dual graviton is simply A ′ M ··· M ,N = A M ··· M ,N , (3.32)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 ≥ A Iµ as the generalized graviphoton In Section 2, we found that the 1-form gauge field A Iµ has a simple structure in terms of thetensors ˆ N and N : A Iµ = ˆ N Iµ + ˆ A kµ ˆ N Ik (M-theory) , A I µ = N I µ + A p µ N Ip (type IIB) . (3.33)In fact, this combination has a clear origin. The basic idea is as follows. Generalized graviphoton in DFT: In type IIB theory, the graviphoton is given by A m µ = g µν g ν m . (3.34)We can consider a generalization of this graviphoton in double field theory (DFT). In DFT,the inverse of the generalized metric has the form, H ˆ I ˆ J = ˆ g MN ˆ g MK ˆ B KN − ˆ B MK ˆ g KN (ˆ g − ˆ B ˆ g − ˆ B ) MN , ( x ˆ I ) ≡ ( x M , ˜ x M ) . (3.35) Note that the B -field in this paper has the opposite sign to the one usually used in DFT. 18e decompose the physical coordinates as ( x M ) = ( x µ , x m ) and define the generalized coor-dinates for the compact directions as ( x I ) ≡ ( x m , ˜ x m ) ( m = 1 , . . . , n − H µI = (cid:0) ˆ g µm , ˆ g µK ˆ B Km (cid:1) = (cid:0) ˆ g µm , ˆ g µν ˆ B νm + ˆ g µp ˆ B pm (cid:1) = ˆ g µν (cid:0) A mν , ˆ B νm + A pν ˆ B pm (cid:1) . (3.36)This leads us to define the generalized graviphoton as A Iµ ≡ g µν H νI = A mµ ˆ B µm + A pµ ˆ B pm (cid:2) g ≡ (ˆ g µν ) − (cid:3) , (3.37)which transforms covariantly under O( n − , n − 1) transformations, and is sometimes usedin the double sigma model (see, e.g., [18, 32]). In the context of DFT, A Mµ has been studiedin [33], where it is called the Kaluza–Klein vector. By using N N I ≡ δ mN ˆ B Nm , (3.38)we observe that the generalized graviphoton can be expressed as A Iµ = N µI + A pµ N pI , (3.39)which has the same structure as (3.33). Generalized graviphoton in EFT: We now consider the case of EFT starting with thegeneralized metric M ˆ I ˆ J in E EFT. Denoting the inverse matrix of M µν by m µν , we definethe generalized graviphoton as A Iµ ≡ m µν M νI . (3.40)In the following, we show that this A Iµ is precisely the 1-form considered in Section 2. To thisend, let us recall that the generalized metric has the structure M IJ = ( L ⊺ ˆ M L ) IJ , M IJ = ( L − ˆ M − L − ⊺ ) IJ . (3.41)By using the fact that the matrix L has a lower-triangular form, we find M µν = ˆ M µν = | ˆ g | g µν ≡ m µν , (3.42) M µI = ( ˆ M − L − ⊺ ) µJ = ˆ M µN ( L − ⊺ ) N J = | ˆ g | g µν (cid:2) ( L − ⊺ ) ν I + A kν ( L − ⊺ ) kI (cid:3) , (3.43)where we have used ( ˆ M ˆ M ˆ N ) = | ˆ g | δ µρ A iρ δ ik g ρσ 00 ˆ G kl δ µσ ˆ A jσ δ jl . (3.44)19hen, we obtain A Iµ = m µν M νI = ( L − ⊺ ) µI + ˆ A kµ ( L − ⊺ ) kI . (3.45)In order to show that this is the same as the 1-form considered in Section 2, let us com-pute the explicit form of ( L − ⊺ ) µI in M-theory/type IIB parameterizations. In the M-theoryparameterization, L is defined as (3.6) and by using the matrix representations of the E generators given in Appendix B, we obtain( L − ⊺ ) ˆ N I ≃ δ i ˆ N ˆ A ˆ Ni i √ ˆ A ˆ Ni ··· i − ˆ A ˆ N [ i i ˆ A i i i √ ˆ A ˆ Ni ··· i ,i − ˆ A ˆ Ni [ i ··· i ˆ A i i i +35 ˆ A ˆ N [ i i ˆ A i i i ˆ A i i i √ ... , (3.46)where i ∈ { i , . . . , i } has been assumed for the fourth row. By using the identification (3.29),( L − ⊺ ) ˆ N I is precisely the same as ˆ N ˆ N I given in (2.44) and the generalized graviphoton is thesame as the 1-form (2.43).On the other hand, in the type IIB parameterization, L is defined as (3.7) and we obtain( L − ⊺ ) N I ≃ δ m µ A αN m A N m m m − ǫ γδ A γN [ m A δ m m √ A αN m ··· m +5 A αN [ m A m ··· m +5 ǫ γδ A γN [ m A δ m m A α m m √ h A N m ··· m , m + ǫ γδ A γN m A δ m ··· m +10 A N [ m m m A m m m m − ǫ γδ A γN [ m A δ m m A m m m m + ǫ αβ ǫ γδ A αN [ m A β m m A γ m m A δ m m i √ ... , (3.47)where m ∈ { m , . . . , m } has been assumed for the fifth row. Again by using the identifica-tion (3.31) and m ∈ { m , . . . , m } , ( L − ⊺ ) N I matches N N I given in (2.48) and the generalizedgraviphoton in the type IIB parameterization is precisely the 1-form (2.47).Here, let us comment on the relation to the series of papers [34–36]. The standard wavesolution in 11D supergravity has the non-vanishing flux associated with the graviphoton A iµ .In [34–36], the wave solution was embedded into EFT, which has non-vanishing A Iµ . Then,by rotating the duality frames, various brane solutions were obtained in EFT. Particularlyin [36], the 1-form A Iµ was regarded as the graviphoton in the (4 + 56)-dimensional exceptionalspace. Since all of their brane solutions in EFT couple to the generalized graviphoton A Iµ ,branes were interpreted as a kind of generalized wave in the exceptional space. Although theexplicit parameterization of A Iµ was not determined there, conceptually, their idea is closelyrelated to the result obtained here. 20 Parameterization of A I p p In this section, we study the parameterization of the higher p -form fields A I p p . A I The 2-form gauge field A I transforms in the string multiplet, characterized by the Dynkinlabel [0 , . . . , , , 0] . It is decomposed as A I µν = ˆ A [ µν ]; i √ ˆ A [ µν ]; i ··· i √ ˆ A [ µν ]; i ··· i ,i ... , A I µν = A αµν √ A [ µν ]; m m √ A α [ µν ]; m ··· m √ A α [ µν ]; m ··· m , m ... , (4.1)and, e.g., the first component in each parameterization can be expanded asˆ A [ µν ]; i = ˆ A µνi + m ˆ A k [ µ ˆ A ν ] ki + m ˆ A k [ µ ˆ A lν ] ˆ A kli , (4.2) A αµν = A αµν + b A p [ µ A αν ] p + b A p [ µ A q ν ] A α pq , (4.3)by introducing parameters m , m , b , and b . We already have the T -duality rules, and byfollowing the same procedure as the 1-form, we can determine these parameters.Repeating the procedure, we find the parameterization A I µν = ˆ N [ µ ; ν ] i + ˆ A k [ µ | ˆ N k ; | ν ] i √ (cid:0) ˆ N [ µ ; ν ] i ··· i + ˆ A k [ µ | ˆ N k ; | ν ] i ··· i (cid:1) √ (cid:0) ˆ N [ µ ; ν ] i ··· i ,i + ˆ A k [ µ | ˆ N k ; | ν ] i ··· i ,i (cid:1) ... , (4.4) A I µν = N α [ µ ; ν ] + A p [ µ | N α p ; | ν ]1 √ (cid:0) N [ µ ; ν ] m m + A p [ µ | N p ; | ν ] m m (cid:1) √ (cid:0) N α [ µ ; ν ] m ··· m + A p [ µ | N α p ; | ν ] m ··· m (cid:1) √ (cid:0) N [ µ ; ν ] m ··· m , m + A p [ µ | N p ; | ν ] m ··· m , m (cid:1) ... . (4.5) Interestingly, the tensors ˆ N and N are precisely the same as those defined in Section 2.4. Theorigin of this simple structure can be understood as follows.For example, let us consider the map (2.38)ˆ A µaα + ˆ A kµ ˆ A kaα M–B = (cid:0) A βµa + A p µ A β p a (cid:1) ǫ βα , (4.6)in which both sides are connected through T -duality. However, the T -duality rule is 9Dcovariant, and even if we replace the index a by the 9D index A = ( µ, a ) , the above relationis still satisfied. Then, choosing A = ν and antisymmetrizing µ and ν , we getˆ A µνα + ˆ A k [ µ ˆ A | k | ν ] α M–B = (cid:0) A βµν + A p [ µ A β | p | ν ] (cid:1) ǫ βα , (4.7)21hich connects the first row of A I µν and the first row of A I µν . In this manner, simply byreplacing an internal index a with an external index ν and acting the antisymmetrization, weobtain 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 in,e.g., [3, 37]. By following the notation of [18], their M-theory parameterizations are A µm = A µm , A µmn = ˆ C µmnβ − A µk ˆ C kmnβ √ , B µνm = ˆ C µνm − A [ µk ˆ C ν ] mk √ , (4.8)while the type IIB parameterizations are A µi = A µi , A µiα = ǫ αβ (cid:0) ˆ C µiβ − A µk ˆ C kiβ (cid:1) , B µν α = C µν α − A [ µj ˆ C | j | ν ] α √ . (4.9)By comparing, e.g., A µmn with B µνm , we find that their results also follow the antisymmetriza-tion rule and seem to be consistent with our results up to conventions. -form and higher p -form Similar to the case of the 2-form, a parameterization of a general p -form can be obtained byacting the antisymmetrization on that of the 1-form. In the case of the 3-form, we obtain A I µνρ = ˆ N [ µ ; νρ ] + ˆ A k [ µ | ˆ N k ; | νρ ]1 √ (cid:0) ˆ N [ µ ; νρ ] i i i + ˆ A k [ µ | ˆ N k ; | νρ ] i i i (cid:1) √ (cid:0) ˆ N [ µ ; νρ ] i ··· i ,i + ˆ A k [ µ | ˆ N k ; | νρ ] i ··· i ,i (cid:1) ... , (4.10) A I µνρ = N [ µ ; νρ ] m + A p [ µ | N p ; | νρ ] m √ (cid:0) N α [ µ ; νρ ] m m m + A p [ µ | N α p ; | νρ ] m m m (cid:1) √ (cid:0) N [ µ ; νρ ] m ··· m , m + A p [ µ | N p ; | νρ ] m ··· m , m (cid:1) ... . (4.11) Compared to the 2-form, the first component in the type IIB side N αµ ; ν has disappeared becausethe number of indices is not enough to account for a 3-form. The 4-form is A I µ ··· µ = √ (cid:0) ˆ N [ µ ; µ ··· µ ] i i + ˆ A k [ µ | ˆ N k ; | µ µ µ ] i i (cid:1) √ (cid:0) ˆ N [ µ ; µ ··· µ ] i ··· i ,i + ˆ A k [ µ | ˆ N k ; | µ µ µ ] i ··· i ,i (cid:1) ... , (4.12) A I µ ··· µ = N [ µ ; µ ··· µ ] + A p [ µ | N p ; | µ µ µ ]1 √ (cid:0) N α [ µ ; µ ··· µ ] m m + A p [ µ | N α p ; | µ µ µ ] m m (cid:1) √ (cid:0) N [ µ ; µ ··· µ ] m m m , m + A p [ µ | N p ; | µ µ µ ] m m m , m (cid:1) ... , (4.13) and higher p -forms are also obtained in a similar manner.We note that if there exists a certain invariant tensor f I p I q I r with symmetry f I p I q I r =( − pq f I q I p I r , we can redefine an r -form A I r r as A I r r → A ′ I r r = A I r r + f I p I q I r A I p p ∧ A I q q . (4.14)In such case, the r -form field is not unique and we cannot fix the parameterization unambigu-ously. 22 Summary and discussion In this paper, we have proposed a systematic way to determine the parameterization of the p -form field A I p 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 beenpartially 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 that donot couple to any supersymmetric branes. However, we have provided another approach todetermine the parameterization of A I p p . We have found that the 1-form field is precisely thegeneralized graviphoton A Iµ = m µν M νI defined by the E generalized metric. By followingthe procedure of [6, 38, 39], we can in principle determine the parameterization of the E generalized metric level by level. We can then determine the full parameterization of the1-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 would be interesting to revisit the worldvolume actions of exoticbranes. In the case of exotic branes, the Wess–Zumino term contains the mixed-symmetrypotentials, but at present, the explicit forms of the brane actions are known for a few examples[31, 40–42]. A manifestly U -duality-covariant Wess–Zumino term, which employs the p -formfields A I p p , has been proposed in [15] and it is important to clarify the connection to the resultsof [31, 40–42] by using the concrete parameterization of A I p p . It would be also interesting todevelop another U -duality-manifest approach to brane actions [17, 43] (see also [16, 18] for asimilar approach).It would also be useful to study the duality transformation rules for more mixed-symmetrypotentials beyond the dual graviton. By following the procedure proposed in this paper, itis a straightforward task to determine such duality rules. Recently, a T -duality manifestformulation for mixed-symmetry potentials has been studied in detail in [44], which aims tobe more useful for determining 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 provemore useful. Acknowledgments The work of JJFM is supported by Plan Propio de Investigaci´on of the University of Murcia R-957/2017 and Fundaci´on S´eneca (21257/PI/19 and 20949/PI/18). The work of YS is supportedby JSPS KAKENHI Grant Numbers 18K13540 and 18H01214.23 Notation In this appendix, we summarize the notation that has been used in this work to denote variousfields 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 respec-tively. 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 ofthe M-theory and type IIB coordinates and the higher/lower-dimensional indices that havebeen used are shown in Figure 1. The D -dimensional coordinates in M-theory and type IIBtheory are denoted by x ˆ M and x M , respectively.In addition, indices for the p -form multiplet are denoted as I p in M-theory and as I p intype IIB theory. In particular, for the 1-form, we denote I ≡ I and I ≡ I . In type IIB theory,the index of the vector representation of the SL(2) S -duality group is represented by α = 1 , U -duality multiplets are considered. Similarly, standardsupergravity fields of M-theory and type II theories and the lower-dimensional fields that ariseafter compactification are considered. B E n generators In this appendix, we show the explicit matrix representation of the E n generators in thevector representation. In the M-theory parameterization, our matrices are consistent with [19].ˆ M M A i m µ a α x ... x d − x d ... x x y x z ← T -duality → S x ... x d − x d ... x x y µ a m A M Figure 1: Left: splittings of the M-theory coordinates ( x ˆ M ) and their index notation. Acompactification on a circle S along the direction x z and a T -duality transformation alongthe x y coordinate are considered. Right: splittings of the type IIB coordinates ( x M ) and theirnotations are shown. A T -duality transformation is taken along the coordinate x y . U -duality-covariant p -form A I p p A I p p –generalized metric M M –generalized vielbein E E –twist matrix L L – D -dim. metric ˆ g g gD -dim. fields ˆ A A , B , C A , B , C D -dim. fields (Section 3) ˆ A A –spacetime metric ˆ g g g internal metric ˆ G G G Table A.1: Summary of the fields that have been used in this work. While the first four linescorrespond to U -duality multiplets, the rest correspond to standard supergravity fields. In thelast two lines we show the d -dimensional fields that appear after compactification. Through the linear map from M-theory parameterization to type IIB parameterization, we alsofind the matrix representations in the type IIB parameterization, which is new.Here, we show the results for E , but the E n generators with n ≤ E generator R i ··· ,i disappears in E because theindex i ranges over seven directions and i ··· automatically vanishes. Conversely, our E generators can be understood as a truncation of the E generators. In E , the matrixrepresentation becomes infinite dimensional, but the first several blocks are the same as the E generators. Accordingly, although we have computed the matrix ( L − ⊺ ) in (3.46) by usingthe E generators, the first four rows do not change even if we use the matrix representationof the E generators. In that sense, the results given in this appendix can be understood asa truncation of the E generators. B.1 M-theory parameterization In the M-theory parameterization, the E n generators are decomposed as { T ˆ α } = (cid:8) K ij , R i , R i ··· , R i ··· ,i , R i , R i ··· , R i ··· ,i , . . . (cid:9) , (B.1) To be more precise, in our matrix representations in M-theory, in the fourth row and below that, we haveused Schouten-like identities; i.e., terms with antisymmetrized nine indices ( · · · ) [ i ··· i ij ] have been droppedbecause they disappear automatically in n ≤ L − ⊺ )in (3.46) because the restriction rule i ∈ { i , . . . , i } has been assumed there and terms with the structure( · · · ) [ i ··· i ij ] disappear even for n = 11 . In this sense, (3.46) can be understood as being obtained from the E generalized metric. n ≤ · · · ) i ··· p ≡ ( · · · ) i ··· i p . (B.2)We are also using the notation δ j ··· j n i ··· i n ≡ n ! δ j ··· j n i ··· i n . (B.3)If we restrict ourselves to the case n ≤ (cid:2) K ij , K kl (cid:3) = δ kj K il − δ il K kj , (B.4) (cid:2) K ij , R k k k (cid:3) = 12! δ k k k jr r R ir r , (B.5) (cid:2) K ij , R k ··· k (cid:3) = 15! δ k ··· k jr ··· r R ir ··· r , (B.6) (cid:2) K ij , R k ··· k ,k (cid:3) = 17! δ k ··· k jr ··· r R ir ··· r ,k + δ kj R k ··· k ,i , (B.7) (cid:2) K ij , R k k k (cid:3) = − δ ir r k k k R jr r , (B.8) (cid:2) K ij , R k ··· k (cid:3) = − δ ir ··· r k ··· k R jr ··· r , (B.9) (cid:2) K ij , R k ··· k ,k (cid:3) = − δ ir ··· r k ··· k R jr ··· r ,k − δ ik R k ··· k ,j , (B.10) (cid:2) R i i i , R j j j (cid:3) = − R i i i j j j , (B.11) (cid:2) R i i i , R j ··· j (cid:3) = − δ j ··· j r ··· r s R i i i r ··· r ,s , (B.12) (cid:2) R i i i , R j j j (cid:3) = 12! δ i i i r r s δ r r tj j j K st − δ i i i j j j δ ts K st , (B.13) (cid:2) R i i i , R j ··· j (cid:3) = 13! δ i i i r r r j ··· j R r r r , (B.14) (cid:2) R i i i , R j ··· j ,j (cid:3) = 15! δ i i i r ··· r j ··· j R r ··· r j , (B.15) (cid:2) R i ··· i , R j j j (cid:3) = 13! δ i ··· i j j j r r r R r r r , (B.16) (cid:2) R i ··· i , R j ··· j (cid:3) = 15! δ i ··· i r ··· r s δ r ··· r tj ··· j K st − δ i ··· i j ··· j δ ts K st , (B.17) (cid:2) R i ··· i , R j ··· j ,j (cid:3) = 12! δ i ··· i r r j ··· j R r r j , (B.18) (cid:2) R i ··· i ,i , R j j j (cid:3) = 15! δ i ··· i j j j r ··· r R r ··· r i , (B.19) (cid:2) R i ··· i ,i , R j ··· j (cid:3) = 12! δ i ··· i j ··· j r r R r r i , (B.20) (cid:2) R i ··· i ,i , R j ··· j ,j (cid:3) = δ i ··· i j ··· j K ij , (B.21) (cid:2) R i i i , R j j j (cid:3) = R i i i j j j , (B.22) If we consider the E algebra, for example (cid:2) R i i i , R j ··· j ,j (cid:3) needs to be modified as (cid:2) R i i i , R j ··· j ,j (cid:3) = (cid:0) δ i i i r ··· r j ··· j R r ··· r j − δ i i i r ··· r [ j ··· j | R r ··· r | j ] (cid:1) . However, the second term on the right-hand side identicallyvanishes for n ≤ n ≤ R i i i , R j ··· j (cid:3) = 15! δ r ··· r sj ··· j R i i i r ··· r ,s . (B.23)We note that our convention will be related to that of [19] as follows:Here K ij R i R i R i ··· R i ··· R i ··· ,i R i ··· ,i [19] K ij R i R i R i ··· − R i ··· R i ··· ,i R i ··· ,i . 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 { x I } = n x i , y i √ , y i ··· √ , y i ··· ,i √ , y i ··· √ , y i ··· ,k √ 8! 3! , y i ··· ,k ··· √ 8! 6! , y i ··· ,k ··· ,i √ 8! 8! o , (B.24)where y [ i ··· ,i ] = 0 . In this paper, in order to reduce the matrix size, we have combined y i ··· ,i and y i ··· , and our y i ··· ,i do not satisfy y [ i ··· ,i ] = 0 . We then find that the following matrices( T ˆ α ) I J satisfy the above E algebra: K pq ≡ diag × − δ iq δ pj δ pri δ j qr √ 2! 2! δ pr ··· i ··· δ j ··· qr ··· √ 5! 5! δ pr ··· i ··· δ j ··· qr ··· δ ji + δ j ··· i ··· δ pi δ jq √ 7! 7! δ pr ··· i ··· δ j ··· qr ··· δ l k + δ j ··· i ··· δ pr k δ l qr √ 8! 3! 8! 3! δ pr ··· i ··· δ j ··· qr ··· δ l ··· k ··· + δ j ··· i ··· δ pr ··· k ··· δ l ··· qr ··· √ 8! 6! 8! 6! δ pr ··· i ··· δ j ··· qr ··· δ l ··· k ··· δ ji + δ j ··· i ··· δ pr ··· k ··· δ l ··· qr ··· δ ji + δ j ··· i ··· δ l ··· k ··· δ pi δ jq √ 8! 8! 8! 8! − δ pq δ IJ − n , (B.25) R p ≡ − δ p ji √ δ j p i ··· √ 5! 2! δ j ··· r i ··· δ p r i − δ j ··· p i ··· i √ 7! 5! δ j ··· ri ··· δ p rs δ s jk − δ j ··· ji ··· δ p k √ 8! 3! 7! δ j ··· i ··· δ l p k ··· √ 8! 6! 8! 3! δ j ··· i ··· δ l ··· r k ··· δ p ir √ 8! 8! 8! 6! , (B.26) R p ≡ − δ ij p √ δ j ··· i p √ 2! 5! δ j ··· i ··· r δ r jp − δ j ··· ji ··· p √ 5! 7! δ j ··· i ··· r δ rs p δ l s i − δ j ··· i ··· i δ l p √ 7! 8! 3! δ j ··· i ··· δ l ··· k p √ 8! 3! 8! 6! 00 0 0 0 0 0 δ j ··· i ··· δ l ··· k ··· r δ jr p √ 8! 6! 8! 8! = ( R p ) T , (B.27)27 p ··· ≡ δ p ··· i ··· j √ δ j ir δ p ··· ri ··· + δ p ··· j i ··· i √ 7! 2! δ j ··· r i ··· δ p ··· k r √ 8! 3! 5! δ rj ··· i ··· δ p ··· jk ··· r + δ j ··· ji ··· δ p ··· k ··· √ 8! 6! 7! − δ j ··· i ··· δ p ··· r k ··· δ l ir √ 8! 8! 8! 3! , (B.28) R p ··· ≡ δ j ··· ip ··· √ δ jri δ j ··· p ··· r + δ j ··· jp ··· i √ 2! 7! δ j ··· i ··· r δ l r p ··· √ 5! 8! 3! δ j ··· ri ··· δ l ··· rp ··· i + δ j ··· i ··· i δ l ··· p ··· √ 7! 8! 6! 00 0 0 0 0 0 − δ j ··· i ··· δ l ··· p ··· r δ jr k √ 8! 3! 8! 8! = ( R p ··· ) T , (B.29) R p ··· ,p ≡ δ p ··· i ··· j δ pi + δ p ··· i ··· i δ pj √ − δ j pk δ p ··· i ··· √ 8! 3! 2! δ j ··· pk ··· δ p ··· i ··· √ 8! 6! 5! δ j ··· pi ··· δ p ··· k ··· δ ji + δ j ··· ji ··· δ p ··· k ··· δ pi √ 8! 8! 7! , (B.30) R p ··· ,p ≡ δ j ··· ip ··· δ jp + δ j ··· jp ··· δ ip √ − δ l i p δ j ··· p ··· √ 2! 8! 3! δ l ··· i ··· p δ j ··· p ··· √ 5! 8! 6! 00 0 0 0 0 0 δ j ··· i ··· p δ l ··· p ··· δ ji + δ j ··· i ··· i δ l ··· p ··· δ jp √ 7! 8! 8! = ( R p ··· ,p ) T . (B.31)We can identify the Cartan generators as { H k } = { K dd − K d +1 d +1 , . . . , K − K zz , K + K + K zz + D } , (B.32)and the positive/negative simple-root generators are { E k } = { K dd +1 , . . . , K z , R z } , { F k } = { K d +1 d , . . . , K z , R z } . (B.33)28hey satisfy the relations[ H k , E l ] = A kl E l , [ H k , F l ] = − A kl F l , [ E k , F l ] = δ kl H l , α n tr( H k H l ) = A kl , α n tr( E k F l ) = δ kl , (B.34)where ( A kl ) = − − − − − − − − , n α n . (B.35)The set of positive/negative root generators can be obtained by taking commutators of thesimple-root generators E k / F k , and they can be summarized as { E α } = { K ij ( i < j ) , R i , R i ··· , R i ··· ,i } , { F α } = { K ij ( i > j ) , R i , R i ··· , R i ··· ,i } . (B.36) B.2 Type IIB parameterization We can transform the E n generators of the M-theory parameterization into the type IIBparameterization by using the linear map (3.14). Namely, we act the following operation tothe matrix representations of the generators: T ( T ˆ α ) IJ ≡ ( S T ) I K ( T ˆ α ) K L S L J . (B.37)Then, T ( T ˆ α ) IJ is the matrix representations in the type IIB parameterization. The explicitform of the constant matrix S I J has been determined such that the algebra of the type IIBgenerators 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-theory29arameterization into those of the type IIB parameterization as follows: T ( K ab ) K ab ' ' T ( K ij ) / / + + * * T (cid:0) K aa − K αα (cid:1) K yy / / K mn T (cid:0) K αβ − δ αβ K γγ (cid:1) ǫ αγ R γβ / / R αβ T ( K αa ) R αa y ' ' T ( R a yz ) K y a B B R α m m T ( R i i i ) / / - - T ( R a a α ) ǫ ⊺ αβ R βa a T ( R a a a ) R a a a y / / R m ··· m T ( R a ··· a yz ) R a ··· a T ( R i ··· i ) / / - - T ( R a ··· a α ) ǫ ⊺ αβ R βa ··· a y / / R α m ··· m T ( R a ··· a ) R a ··· a y , y ' ' T ( R i ··· i ,i ) / / - - T ( R a ··· a yz,α ) ǫ ⊺ αβ R βa ··· a = = R m ··· m , m T ( R a ··· a yz,a ) R a ··· a y ,a (B.38)A similar map for the positive-level generators can be found by taking the generalized trans-pose; e.g., T ( R a a a ) = (cid:2) T ( R a a a ) (cid:3) T = (cid:2) R a a a y (cid:3) T = R a a a y . (B.39)We then obtain the E n generators ( n ≤ 8) in the type IIB parameterization: { T α } = { K mn , R αβ , R m α , R m ··· , R m ··· α , R m ··· , m R α m , R m ··· , R α m ··· , R m ··· , m } . (B.40)By using the notations δ n ··· n n m ··· m n ≡ n ! δ n ··· n n m ··· m n , ( · · · ) m ··· p ≡ ( · · · ) m ··· m p , δ αβγδ ≡ δ ( α ( γ δ β ) δ ) , (B.41)30heir matrix representations are found as follows: K rs ≡ diag × − δ ms δ rn δ αβ δ rm δ ns δ rt m δ n st √ 3! 3! δ αβ δ rt ··· m ··· δ n ··· st ··· √ 5! 5! δ rt ··· m ··· δ n ··· st ··· δ nm + δ n ··· m ··· δ rm δ ns √ 6! 6! δ α β δ rt ··· m ··· δ n ··· st ··· √ 7! 7! δ αβ ( δ rt ··· m ··· δ n ··· st ··· δ q p + δ n ··· m ··· δ rtp δ q st ) √ 7! 2! 7! 2! δ rt ··· m ··· δ n ··· st ··· δ p ··· q ··· + δ n ··· m ··· δ rt p ··· δ q ··· st √ 7! 4! 7! 4! δ αβ ( δ rt ··· m ··· δ n ··· st ··· δ q ··· p ··· + δ n ··· m ··· δ rt ··· p ··· δ q ··· st ··· ) √ 7! 6! 7! 6! δ rt ··· m ··· δ n ··· st ··· δ q ··· p ··· δ nm + δ n ··· m ··· δ rt ··· p ··· δ q ··· st ··· δ nm + δ n ··· m ··· δ q ··· p ··· δ rm δ ns √ 7! 7! 7! 7! − δ rs δ IJ − n , (B.42) R γδ ≡ δ α ( γ ǫ δ ) β δ nm δ α ( γ ǫ δ ) β δ n ··· m ··· √ 5! 5! ( R γδ ) α β δ n ··· m ··· √ 7! 7! δ α ( γ ǫ δ ) β δ n ··· m ··· δ q p √ 7! 2! 7! 2! δ α ( γ ǫ δ ) β δ n ··· m ··· δ q ··· p ··· √ 7! 6! 7! 6! 00 0 0 0 0 0 0 0 0 0 h ( R γδ ) α β ≡ δ α α β ( γ ǫ δ ) β + δ α α β ( γ ǫ δ ) β i , (B.43) R r γ ≡ δ αγ δ r mn ǫ βγ δ nr m √ δ αγ δ n r m ··· √ 5! 3! ǫ βγ h c δ n ··· r m ··· m + δ n ··· tm ··· δ r mt i √ 6! 5! δ α βγ δ n ··· r m ··· √ 7! 5! δ αγ h c δ n ··· nm ··· δ r p + δ n ··· sm ··· δ r st δ tnp i √ 7! 2! 6! δ α ( β ǫ β γ δ n ··· m ··· δ r p √ 7! 2! 7! ǫ βγ δ n ··· m ··· δ q r p ··· √ 7! 4! 7! 2! δ αγ δ n ··· m ··· δ p ··· q r p ··· √ 7! 6! 7! 4! ǫ βγ δ n ··· m ··· δ q ··· tp ··· δ r mt √ 7! 7! 7! 6! (cid:20) c ≡ √ (cid:21) , (B.44)31 γ r ≡ δ γβ δ nmr ǫ αγ δ n mr √ δ γβ δ n ··· m r √ 3! 5! ǫ αγ h c δ n ··· nm ··· r + δ n ··· m ··· t δ ntr i √ 5! 6! δ αγβ δ n ··· m ··· r √ 5! 7! δ γβ h c δ n ··· m ··· m δ q r + δ n ··· m ··· s δ str δ q tm i √ 6! 7! 2! δ ( α β ǫ α γ δ n ··· m ··· δ q r √ 7! 7! 2! ǫ αγ δ n ··· m ··· δ q ··· p r √ 7! 2! 7! 4! δ γβ δ n ··· m ··· δ q ··· p ··· p r √ 7! 4! 7! 6! 00 0 0 0 0 0 0 0 0 ǫ αγ δ n ··· m ··· δ q ··· p ··· t δ ntr √ 7! 6! 7! 7! = ( R r γ ) T , (B.45) R r ··· ≡ δ r ··· m n √ − δ αβ δ nr ··· m ··· √ h δ n t m ··· δ r ··· mt + c δ n r ··· m ··· m i √ 6! 3! δ αβ δ n ··· t m ··· δ r ··· t p √ 7! 2! 5! h δ n ··· sm ··· δ t np ··· δ r ··· st + c δ n ··· nm ··· δ r ··· p ··· i √ 7! 4! 6! − δ αβ δ n ··· m ··· δ q r ··· p ··· √ 7! 6! 7! 2! δ n ··· m ··· δ q ··· pt δ t r ··· p ··· √ 7! 7! 7! 4! (cid:20) c ≡ √ (cid:21) , (B.46) R r ··· ≡ δ n mr ··· √ − δ αβ δ n ··· mr ··· √ h δ n ··· m t δ nt r ··· + c δ n ··· nm r ··· i √ 3! 6! δ αβ δ n ··· m ··· t δ t q r ··· √ 5! 7! 2! h δ n ··· m ··· s δ q ··· t m δ st r ··· + c δ n ··· m ··· m δ q ··· r ··· i √ 6! 7! 4! − δ αβ δ n ··· m ··· δ q ··· p r ··· √ 7! 2! 7! 6! 00 0 0 0 0 0 0 0 0 δ n ··· m ··· δ qt p ··· δ q ··· t r ··· √ 7! 4! 7! 7! = ( R r ··· ) T , (B.47)32 r ··· γ ≡ δ αγ δ r ··· m ··· n √ ǫ βγ h c δ nr ··· m ··· m − δ r ··· m ··· δ nm i √ − δ α βγ δ nr ··· m ··· √ − δ αγ δ r ··· tm ··· δ n p t √ 7! 2! 3! ǫ βγ δ r ··· tm ··· δ n ··· p ··· t √ 7! 4! 5! − δ αγ h c δ n ··· nm ··· δ r ··· p ··· − δ n ··· p ··· δ nr ··· m ··· i √ 7! 6! 6! δ α ( β ǫ β γ δ n ··· m ··· δ r ··· p ··· √ 7! 7! 6! ǫ βγ δ n ··· m ··· δ r ··· tp ··· δ q mt √ 7! 7! 7! 2! h c ≡ − √ i , (B.48) R γ r ··· ≡ δ γβ δ n ··· mr ··· √ ǫ αγ h c δ n ··· nmr ··· − δ n ··· r ··· δ nm i √ − δ αγβ δ n ··· mr ··· √ − δ γβ δ n ··· r ··· t δ q tm √ 3! 7! 2! ǫ αγ δ n ··· r ··· t δ q ··· tm ··· √ 5! 7! 4! − δ γβ h c δ n ··· m ··· m δ q ··· r ··· − δ q ··· m ··· δ n ··· mr ··· i √ 6! 7! 6! 00 0 0 0 0 0 0 0 δ ( α β ǫ α γ δ n ··· m ··· δ q ··· r ··· √ 7! 7! 6! 00 0 0 0 0 0 0 0 0 ǫ αγ δ n ··· m ··· δ q ··· r ··· t δ ntp √ 7! 2! 7! 7! = ( R r ··· γ ) T , (B.49) R r ··· , r ≡ h c , δ rn δ r ··· m ··· m − δ r ··· nm ··· δ rm i √ δ αβ δ nrp δ r ··· m ··· √ 7! 2! δ n rp ··· δ r ··· m ··· √ 7! 4! 3! δ αβ δ n ··· rp ··· δ r ··· m ··· √ 7! 6! 5! h c , δ n ··· nm ··· δ r ··· p ··· δ rm − δ n ··· rm ··· δ r ··· p ··· δ nm i √ 7! 7! 6! (cid:20) c , ≡ √ √ (cid:21) , (B.50)33 r ··· , r ≡ h c , δ mr δ n ··· nr ··· − δ mn ··· r ··· δ nr i √ δ αβ δ q mr δ n ··· r ··· √ 7! 2! δ q ··· m r δ n ··· r ··· √ 3! 7! 4! δ αβ δ q ··· m ··· r δ n ··· r ··· √ 5! 7! 6! 00 0 0 0 0 0 0 0 0 h c , δ n ··· m ··· m δ q ··· r ··· δ nr − δ n ··· m ··· r δ q ··· r ··· δ nm i √ 6! 7! 7! = ( R r ··· , r ) T . (B.51)Again, we can identify the Cartan generators as { H k } = { K dd − K d +1 d +1 , . . . , K − K , K + K − D − R , R , K − K } , (B.52)and the positive/negative simple-root generators are { E k } = { K dd +1 , . . . , K , R , R , K } , { F k } = { K d +1 d , . . . , K , R , − R , K } . (B.53)The set of positive/negative root generators can be summarized as { E α } = { K mn ( m < n ) , R , R m α , R m ··· , R m ··· α , R m ··· , m } , { F α } = { K mn ( m > n ) , − R , R α m , R m ··· , R α m ··· , R m ··· , m } . (B.54)We have checked that the obtained type IIB generators satisfy the following E algebra: (cid:2) K mn , K pq (cid:3) = δ pn K mq − δ mq K pn , (B.55) (cid:2) K mn , R p α (cid:3) = δ p nr R mr α , (B.56) (cid:2) K mn , R p ··· (cid:3) = 13! δ p ··· nr R mr , (B.57) (cid:2) K mn , R p ··· α (cid:3) = 15! δ p ··· nr ··· R mr ··· α , (B.58) (cid:2) K mn , R p ··· , p (cid:3) = 16! δ p ··· nr ··· R mr ··· , p + δ pn R p ··· , m , (B.59) (cid:2) K mn , R α p (cid:3) = − δ mrp R α nr , (B.60) (cid:2) K mn , R p ··· (cid:3) = − δ mr p ··· R nr , (B.61) (cid:2) K mn , R α p ··· (cid:3) = − δ mr ··· p ··· R α nr ··· , (B.62) (cid:2) K mn , R p ··· , p (cid:3) = − δ mr ··· p ··· R nr ··· , p − δ mp R p ··· , n , (B.63) (cid:2) R αβ , R γδ (cid:3) = δ σ ( α ǫ β ) γ R σδ + δ σ ( α ǫ β ) δ R γσ , (B.64)34 R αβ , R m γ (cid:3) = δ σ ( α ǫ β ) γ R m σ , (B.65) (cid:2) R αβ , R m ··· γ (cid:3) = δ σ ( α ǫ β ) γ R m ··· σ , (B.66) (cid:2) R αβ , R γ m (cid:3) = − δ γ ( α ǫ β ) σ R σ m , (B.67) (cid:2) R αβ , R γ m ··· (cid:3) = − δ γ ( α ǫ β ) σ R σ m ··· , (B.68) (cid:2) R m α , R n β (cid:3) = − ǫ αβ R m n , (B.69) (cid:2) R m α , R n ··· (cid:3) = R m n ··· α , (B.70) (cid:2) R m α , R n ··· β (cid:3) = − ǫ αβ δ n ··· r ··· s R m r ··· , s , (B.71) (cid:2) R m α , R β n (cid:3) = δ βα δ m pr δ qrn K pq − δ βα δ m n δ qp K pq − ǫ βγ δ m n R αγ , (B.72) (cid:2) R m α , R n ··· (cid:3) = 12! ǫ αβ δ m r n ··· R β r , (B.73) (cid:2) R m α , R β n ··· (cid:3) = − δ βα δ m r ··· n ··· R r ··· , (B.74) (cid:2) R m α , R n ··· , n (cid:3) = 15! ǫ αβ δ m r ··· n ··· R β r ··· n , (B.75) (cid:2) R m ··· , R n ··· (cid:3) = 13! δ n ··· r s R m ··· r , s , (B.76) (cid:2) R m ··· , R α n (cid:3) = 12! ǫ αβ δ m ··· n r R r β , (B.77) (cid:2) R m ··· , R n ··· (cid:3) = 13! δ m ··· pr δ qr n ··· K pq − δ m ··· n ··· δ qp K pq (B.78) (cid:2) R m ··· , R α n ··· (cid:3) = 12! δ m ··· r n ··· R α r , (B.79) (cid:2) R m ··· , R n ··· , n (cid:3) = − δ m ··· r n ··· R r n (B.80) (cid:2) R m ··· α , R β n (cid:3) = − δ βα δ m ··· n r ··· R r ··· , (B.81) (cid:2) R m ··· α , R n ··· (cid:3) = 12! δ m ··· n ··· r R r α , (B.82) (cid:2) R m ··· α , R β n ··· (cid:3) = 15! δ βα δ m ··· pr ··· δ qr ··· n ··· K pq − δ βα δ m ··· n ··· δ qp K pq − ǫ βγ δ m ··· n ··· R αγ , (B.83) (cid:2) R m ··· α , R n ··· , n (cid:3) = − ǫ αβ δ m ··· rn ··· R β rn , (B.84) (cid:2) R m ··· , m , R α n (cid:3) = 15! ǫ αβ δ m ··· n r ··· R r ··· n β , (B.85) (cid:2) R m ··· , m , R n ··· (cid:3) = − δ m ··· n ··· r R r m (B.86) (cid:2) R m ··· , m , R α n ··· (cid:3) = − ǫ αβ δ m ··· n ··· r R rm β , (B.87) (cid:2) R m ··· , m , R n ··· , n (cid:3) = δ m ··· n ··· K mn , (B.88) (cid:2) R α m , R β n (cid:3) = ǫ αβ R m n , (B.89) (cid:2) R α m , R n ··· (cid:3) = − R α m n ··· , (B.90) (cid:2) R α m , R β n ··· (cid:3) = 15! ǫ αβ δ r ··· sn ··· R m r ··· , s , (B.91) (cid:2) R m ··· , R n ··· (cid:3) = − δ r sn ··· R m ··· r , s . 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