E 6(6) Exceptional Drinfel'd Algebras
EE Exceptional Drinfel’d Algebras
Emanuel Malek a , Yuho Sakatani b and Daniel C. Thompson c,d a Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut),Am M¨uhlenberg 1, 14476 Potsdam, Germany b Department of Physics, Kyoto Prefectural University of Medicine,Kyoto 606-0823, Japan c Theoretische Natuurkunde, Vrije Universiteit Brussel, and the International SolvayInstitutes,Pleinlaan 2, B-1050 Brussels, Belgium d Department of Physics, Swansea University,Swansea SA2 8PP, United Kingdom
Abstract
The exceptional Drinfel’d algebra (EDA) is a Leibniz algebra introduced to providean algebraic underpinning with which to explore generalised notions of U-duality in M-theory. In essence, it provides an M-theoretic analogue of the way a Drinfel’d double encodesgeneralised T-dualities of strings. In this note we detail the construction of the EDA in thecase where the regular U-duality group is E . We show how the EDA can be realisedgeometrically as a generalised Leibniz parallelisation of the exceptional generalised tangentbundle for a six-dimensional group manifold G , endowed with a Nambu-Lie structure. Whenthe EDA is of coboundary type, we show how a natural generalisation of the classical Yang-Baxter equation arises. The construction is illustrated with a selection of examples includingsome which embed Drinfel’d doubles and others that are not of this type. [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] J u l ontents E EDA 3 E EDA from generalised frame fields 9 ρ -twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Nambu 3- and 6-brackets from ρ -twisting . . . . . . . . . . . . . . . . . . . . . . 134.3 The generalised YB equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 r -matrix EDAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.4 Explicit examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4.1 Trivial non-examples based on SO ( p, q ) . . . . . . . . . . . . . . . . . . . 185.4.2 An example both coboundary and non-coboundary solutions . . . . . . . 185.4.3 An example with ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4.4 An r -matrix EDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 E × R + Algebra and conventions 21 Introduction
Dualities play an important role in our understanding of string theory. One of the best-understood dualities is T-duality, which relates string theory on backgrounds with U (1) d isome-tries, with the backgrounds related by O( d, d ) transformations. These T-dualities are alreadyvisible in perturbative string theory, and are enlarged into U-dualities in the non-perturbativeframework of M-theory [1, 2]. Generalisations of Abelian T-dualities exist for backgrounds withnon-Abelian isometries, leading to non-Abelian T-duality (NATD) [3], and for backgroundswithout any isometries, called Poisson-Lie T-duality (PLTD) [4, 5]. Instead of the isometryalgebra, PLTD is controlled by an underlying Drinfel’d double.Unlike Abelian T-duality, which is an equivalence between string theories on different back-grounds to all orders in the string coupling and string length, these generalised T-duality arecurrently best understood at the supergravity level and only to a limited extend beyond leadingorder in α (cid:48) [6] and their status as true dualities of the string genus expansion remains doubt-ful [7]. Nonetheless, NATD and PLTD have led to fruitful results. For example, NATD has beensuccessfully used as solution-generating mechanisms of supergravity [8], leading to the discoveryof new minimally supersymmetric AdS backgrounds starting with [9] (see [10] for a review andfurther references). Moreover, there is a close connection between PLTD and the (modified)classical Yang-Baxter equation which controls integrable deformations of σ -models [11, 12].The non-perturbative generalisation of Poisson-Lie T-duality to a U-duality version in M-theory, or more conservatively as a solution-generating mechanism of 11-dimensional supergrav-ity, has long been an open problem, which was recently addressed in [13, 14] and further elabo-rated on in [15–17]. Building on the interpretation of PLTD and Drinfel’d doubles within DoubleField Theory (DFT), [18–20], [13, 14] used Exceptional Field Theory (ExFT)/Exceptional Gen-eralised Geometry to propose a natural generalisation of the Drinfel’d double for dualities alongfour spacetime dimensions. This “Exceptional Drinfel’d Algebra” (EDA) was shown to leadto a new solution-generating mechanism of 11-dimensional supergravity that suggests a no-tion of Poisson-Lie U-duality, as well as a generalisation of the classical Yang-Baxter equation.Other recent works [21–23] have considered closely related ideas, although the detailed relationbetween these approaches and the EDA is not completely apparent.In this paper, we will further develop the ideas of [13,14] by constructing EDAs and Poisson-Lie U-duality amongst six directions. We choose six dimensions, because important new featuresarise when dualities are considered in six directions. This is because now the 6-form can com-pletely wrap the six directions we are considering. As a result, the e algebra contains agenerator, corresponding to a hexavector, which will generate new kinds of dualities and defor-mations which have no counterpart in PLTD, as we will see.The outline of the rest of this paper is as follows: in section 2 we describe the EDA froma purely algebraic perspective. In section 3 we show how the EDA can be realised withinexceptional generalised geometry as a Leibniz parallelisation of a particular type of group man-ifold G , that we will call a (3 , E EDA
Before specialising to the case of E we begin by presenting some generalities of the Excep-tional Drinfel’d Algebra. The EDA, d n , is a Leibniz algebra which is a subalgebra of E n ( n )1 , admitting a “maximally isotropic” subalgebra, as we will define shortly. In table 1 we pro-vide details of the representations of E n ( n ) inherited from the exceptional field theory (ExFT)approach to eleven-dimensional supergravity that are useful to the present construction. D E n ( n ) H n R R R R / Z
10 5 5 10 ,
5) USp(4) × USp(4) / Z
16 10 16 45
USp(8) / Z
27 27 78 351 (cid:48)
Table 1:
The split real form of exceptional groups E n ( n ) with D = 11 − n , their maximalcompact subgroups H n and representations R . . . R appearing in the tensor hierarchy ofExFT. In this work we will be mostly concerned with representations R and R which willbe associated to the generalised tangent bundles E and N respectively. We denote the generators of d n by { T A } , with the index A inherited from the ¯ R represen-tation of E n ( n ) ExFT and their product by T A ◦ T B = X ABC T C , (2.1)with X ABC structure constants which are not necessarily antisymmetric in their lower indices.The product obeys the Leibniz identity, namely T A ◦ ( T B ◦ T C ) = ( T A ◦ T B ) ◦ T C + T B ◦ ( T A ◦ T C ) , (2.2)which implies for the structure constants X AC D X BDE − X BC D X ADE + X ABD X DC E = 0 . (2.3)Note that if the Leibniz algebra is a Lie algebra, i.e. the X ABC are antisymmetric in their lowerindices, then this reduces to the Jacobi identity.We place two further (linear) requirements on the EDA. Firstly, we demand that there is a In general one can allow for EDAs as subalgebras of E n ( n ) × R + , see [14]. However, we will not deal with theextra R + factor here. g spanned by { T a } ⊂ { T A } obeying g ⊗ g | ¯ R = 0 , (2.4)in which the representation ¯ R is found in table 1. We call such a subalgebra g maximallyisotropic. We will be interested here in the case that dim g = n as this is relevant to the M-theory context. Since G = exp g acts adjointly on d n , it follows that G should be endowedwith a trivector and hexavector. We will further require that these objects give rise to a 3- and6-bracket on g ∗ , thereby imposing some further restrictions on the structure constants X ABC . These additional requirements imply that the EDA can be given a geometrical realisation interms of certain generalised frames whose action is mediated by the generalised Lie derivative(3.4), as we will show in section 3.Let us now discuss these restrictions in detail.
We now study in detail the consequence of the requirements of the maximally isotropic subalge-bra g and its adjoint action. Since these constraints arise from placing requirements directly tothe form of X ABC we describe them as linear constraints; this is to be contrasted with quadraticconstraints of the form X = 0 that arise from the Leibniz identity.Firstly, since g is a Lie algebra, we immediately have T a ◦ T b = f abc T c , (2.5)with f abc antisymmetric in a , b . Secondly, the adjoint action of g ∈ G = exp g on d n impliesthat g · T A · g − = ( A g ) A B T B , (2.6)with ( A g ) A B ∈ E since g ⊂ d n ⊂ e . Let us denote the adjoint action of g ∈ G on g by a g .Then g · T A · g − takes the form: g · T a · g − = ( a g ) ab T b ,g · T a a · g − = − λ a a cg ( a g ) cb T b + ( a − g ) b a ( a − g ) b a T b b ,g · T a ...a · g − = (cid:0) λ a ...a cg + 5 λ [ a a a g λ a a ] cg (cid:1) ( a g ) cb T b − λ [ a a a g ( a − g ) b a ( a − g ) b a ] T b b + ( a − g ) b [ a . . . ( a − g ) b a ] T b ...b . (2.7) There is another inequivalent way to maximally solve the condition eq. (2.4) with dim g = n − It is worth emphasising that these are impositions beyond simply demanding that g be a maximal isotropic. To be more precise we inherit an action via the rack product: g · T A · g − ≡ g − (cid:46) T A ≡ T A + h ◦ T A + 12 h ◦ ( h ◦ T A ) + · · · ( g − ≡ e h ) . G admits a totally antisymmetric trivector λ abc and totally antisymmetric hexavector λ a ...a which control its adjoint action on the generators T ab and T a ...a .Equations (2.7) imply that ( λ g ) abc and ( λ g ) a ...a vanish at the identity, i.e.( λ e ) abc = ( λ e ) a ...a = 0 , (2.8)and they inherit a group composition rule λ a a a hg = λ a a a g + ( a − g ) c a ( a − g ) c a ( a − g ) c a λ c c c h ,λ a ...a hg = λ a ...a g + ( a − g ) c a . . . ( a − g ) c a λ c ...c h + 10 λ [ a a a g ( a − g ) c a ( a − g ) c a ( a − g ) c a ] λ c c c h , (2.9)for g, h ∈ G .Finally, we come to the second condition on the EDA, i.e. the existence of a 3- and 6-bracketon g ∗ . This is equivalent to imposing the following differential conditions on λ abc and λ a ...a : dλ a a a = r b (cid:16) f ba a a + 3 f bc [ a λ | c | a a ] (cid:17) ,dλ a ...a = r b (cid:16) f ba ...a + 6 f bc [ a λ | c | a ...a ] + 10 f b [ a a a λ a a a ] (cid:17) , (2.10)where r = r a T a are the right-invariant 1-forms on G obeying dr a = f bca r b ∧ r c and we havedropped the subscript g on λ (3) and λ (6) . The f ba ...a and f ba ...a are structure constants fora 3- and 6-bracket and are totally antisymmetric in their upper indices. In fact, as we will seein section 2.2, the Leibniz identity implies further properties of the trivector and hexavector,in particular that they define a certain Nambu 3- and 6-bracket which are compatible with theLie bracket on G . Therefore, it seems apt to call G a (3,6)-Nambu-Lie Group.With the above conditions, the EDA takes the following form T a ◦ T b = f abc T c ,T a ◦ T b b = f ab b c T c + 2 f ac [ b T b ] c ,T a ◦ T b ...b = − f ab ...b c T c + 10 f a [ b b b T b b ] − f ac [ b T b ...b ] c ,T a a ◦ T b = − f ba a c T c + 3 f [ c c [ a δ a ] b ] T c c ,T a a ◦ T b b = − f ca a [ b T b ] c + f c c [ a T a ] b b c c ,T a a ◦ T b ...b = 5 f ca a [ b T b ...b ] c ,T a ...a ◦ T b = f ba ...a c T c − f b [ a a a T a a ] − f c [ a a a δ a b T a ] c + 5 f bc [ a T a ...a ] c + 10 f c c [ a δ a b T a a a ] c c ,T a ...a ◦ T b b = 2 f ca ...a [ b T b ] c − f c [ a a a T a a ] b b c ,T a ...a ◦ T b ...b = − f ca ...a [ b T b ...b ] c . (2.11)5 .2 Leibniz identity constraints We will now study the compatibility conditions between the Lie algebra g , the 3-bracket and6-bracket, as well as their appropriate “closure” conditions that are required for the EDA tosatisfy the Leibniz identity of eq. (2.2). This yields a number of immediate constraints. Inparticular, we obtain the following fundamental identities, i.e. generalisations of Jacobi forhigher brackets, 0 = 3 f [ a a c f a ] cb , (2.12)0 = f adc c f db b b − f dc c [ b f ab b ] d + f d d [ c f ac ] b b b d d , (2.13)0 = f adc ...c f db ...b − f dc ...c [ b f ab ...b ] d , (2.14)as well as compatibility conditions between the dual structure constants f ba a a , f ba ...a andthe Lie algebra structure constants. These compatibility conditions take the form of cocycleconditions 0 = f a a c f cb b b + 6 f c [ a [ b | f a ] c | b b ] , (2.15)0 = f a a c f cb ...b + 12 f c [ a [ b | f a ] c | b ...b ] − f [ a [ b b b f a ] b b b ] , (2.16)as well as the additional constraint f d d a f cd d b = 0 . (2.17)If we only consider EDAs d n with n ≤
6, as we are doing here, the conditions given bythe above eqs. (2.12)-(2.17) are equivalent to imposing the Leibniz identity. This is because in n ≤
6, the fundamental identity for the six-bracket implies that f ba ...a = 0. However, since thestructure we are studying here will also exist for n >
6, we will keep the remaining discussion asdimension-independent as possible, whilst keeping in mind that for n >
6, the Leibniz identitywill lead to further or modified compatibility conditions between f abc , f ba a a and f ba ...a .These additional constraints will need to be studied using EDAs based on E and higher.Before interpreting these constraints, we remark that the Leibniz identity ensures, much asthe structure constants of a Lie algebra g are invariant under G = exp g acting adjointly, thatthe EDA structure constants enjoy an invariance X ABD ( A g ) DC = ( A g ) AD ( A g ) BE X DEC . (2.18)Substitution of eq. (2.7) here results in a variety of identities that we shall revisit later on.6 .2.1 Fundamental identities Let us now introduce the 3-bracket { } and 6-bracket { } on g ∗ with structure constants f ba a a and f ba ...a , respectively, i.e. { x, y, z } = f ab b b x b y b z b , { u, v, w, x, y, z } = f ab ...b u b v b w b x b y b z b . (2.19)The conditions (2.12) and (2.13) imply that the 3- and 6-brackets satisfy { x , x , { x , x , x } } = {{ x , x , x } , x , x } + { x , { x , x , x } , x } + { x , x , { x , x , x } } − { ∆( x ) , x , x , x , x } + { ∆( x ) , x , x , x , x } , { x , . . . , x , { y , . . . , y } } = {{ x , . . . , x , y } , y , . . . , y } + { y , { x , . . . , x , y } , y , . . . , y } + { y , y , { x , . . . , x , y } , y , . . . , y } + { y , . . . , y , { x , . . . , x , y } , y , y } + { y , . . . , y , { x , . . . , x , y } , y } + { y , . . . , y , { x , . . . , x , y } } , (2.20)for all x , . . . , x , y , . . . , y ∈ g ∗ , and where we used the Lie bracket on g to define the ad-invariant co-product ∆ on g ∗ ∆ : g ∗ −→ g ∗ ∧ g ∗ ,ad x ∆( y ) = ∆( ad x y ) , ∀ x ∈ g , y ∈ g ∗ , (2.21)which is given, assuming a basis { T a } for g ∗ , by∆( x a T a ) = 12 f bca x a T b ∧ T c . (2.22)We see that the 6-bracket must satisfy the fundamental identity for Nambu 6-brackets, whilethe 3-bracket’s fundamental identity is modified by the 6-bracket and the co-product definedby the structure constants of g . The first set of compatibility conditions, eqs. (2.15) and (2.16), between the 3- and 6-bracketsand the Lie algebra g imply that f ba a a defines a g -cocycle and that f ba ...a is an f -twisted g -cocycle, as follows. f ba a a and f ba ...a define Λ g - and Λ g -valued 1-cochains f : g −→ Λ g ,f : g −→ Λ g , (2.23)7efined by f ( x ) = 13! x b f ba a a T a ∧ T a ∧ T a , ∀ x = x a T a ∈ g ,f ( x ) = 16! x b f ba ...a T a ∧ . . . ∧ T a , ∀ x = x a T a ∈ g . (2.24)Using the coboundary operator d : g ∗ ⊗ Λ p g −→ Λ g ∗ ⊗ Λ p g , for p = 3 and p = 6 here, df ( x, y ) ≡ ad x f ( y ) − ad y f ( x ) − f ([ x, y ]) , (2.25) df ( x, y ) ≡ ad x f ( y ) − ad y f ( x ) − f ([ x, y ]) , (2.26)the conditions (2.15) and (2.16) are more elegantly stated as df ( x, y ) = 0 , df ( x, y ) + f ( x ) ∧ f ( y ) = 0 . (2.27)The coboundary operator is nilpotent with d : Λ p g −→ g ∗ ⊗ Λ p g defined as dρ p ( x ) = ad x ρ p , (2.28)for all x ∈ g and ρ p ∈ Λ p g . Therefore, the cocycle conditions (2.27) can be solved by the(twisted) coboundaries f = dρ , f = dρ + 12 ρ ∧ dρ . (2.29)In components, these are equivalent to f ab b b = 3 f ac [ b ρ | c | b b ] ,f ab ...b = 6 f ac [ b | ρ c | b ...b ] + 30 f ac [ b ρ | c | b b ρ b b b ] . (2.30)The coboundary case is related to a generalisation of Yang-Baxter deformations. The trivector ρ a a a and the hexavector ρ a ...a correspond to the M-theoretic analogue of the classical r -matrix. The equations corresponding to the classical Yang-Baxter equations for the r -matricesare implied by substituting the solutions (2.30) to the fundamental identities (2.13) and (2.14).We will discuss this further in section 4.Finally, the additional constraint (2.17) implies that the ad-invariant co-product ∆ on g ∗ (2.21) defines a commuting subspace of the 3-bracket: { ∆( x ) , x } = 0 , ∀ x , x ∈ g ∗ . (2.31)8 E EDA from generalised frame fields
We now provide a geometric realisation of the E EDA by constructing a Leibniz parallelisa-tion [26–31] of the exceptional generalised tangent bundle [32–38] E ∼ = T M ⊕ Λ T ∗ M ⊕ Λ T ∗ M , (3.1)in which we identify the manifold M = G = exp g . We will also be interested in a second bundle N ∼ = T ∗ M ⊕ Λ T ∗ M ⊕ ( T ∗ M ⊗ Λ T ∗ M ) . (3.2)The action of sections of these bundles, V = v + ν + ν ∈ Γ( E ) , W = w + ω + ω ∈ Γ( E ) , X = χ + χ + χ , ∈ Γ( N ) , (3.3)is mediated by the generalised Lie derivative [34–36] defined as L V W = [ v, w ] + (cid:0) L v ω − ı w dν (cid:1) + (cid:0) L v ω − ı w dν − ω ∧ dν (cid:1) , (3.4) L V X = L v χ + (cid:0) L v χ − χ ∧ dν (cid:1) + (cid:0) L v χ , + jχ ∧ dν + jχ ∧ dν (cid:1) . (3.5)We define [35] a symmetric bilinear map (cid:104)· , ·(cid:105) : E × E → N as (cid:104) V, W (cid:105) = ( ı v ω + ı w ν ) + ( ı v ω − ν ∧ ω + ı w ν ) + ( jν ∧ ω + jω ∧ ν ) , (3.6)such that the generalized Lie derivative satisfies (cid:104)L U V, W (cid:105) + (cid:104) V, L U W (cid:105) = L U (cid:104) V, W (cid:105) ∀
U, V, W ∈ Γ( E ) . (3.7)The parallelisation consists of a set of sections E A ∈ Γ( E ) that: • form a globally defined basis for Γ( E ) • give rise to an E element , E AM , whose matrix entries are the components of E A • realise the algebra of the EDA through the generalised Lie derivative L E A E B = − X ABC E C , (3.8)where the constants X ABC are the same as those defined through the relations in eq.(2.11) and obey the Leibniz identity.The parallelisation can be directly constructed in terms of the right-invariant Maurer-Cartanone-forms on G , r a , their dual vector fields e a , and the trivector, λ a a a , and hexavector, λ a ...a .This can thus be thought of as a special example of the more general prescription of [39], where An extension of this setup is to allow E AM to be elements of E × R + , though for simplicity in thepresentation we shall demand no R + weighting.
9e only make use of the aforementioned geometric data on the (3 , G .Following the decomposition of EDA generators we write E A = { E a , E a a , E a ...a } with E a = e a , E a a = − λ a a b e b + r a ∧ r a ,E a ...a = (cid:0) λ a ...a b + 5 λ [ a a a λ a a ] b (cid:1) e b − λ [ a a a r a ∧ r a ] + r a ∧ . . . ∧ r a . (3.9)It is straightforward, but indeed quite lengthy, to verify that these furnish the EDA algebra.A first check is to see that after using the identities (2.10) to evaluate derivatives we can go tothe identity of M where λ a a a and λ a ...a vanish. One then has to use the adjoint invarianceconditions that follow from eq (2.18) to conclude that this holds away from the identity.If we specialise now to the case of f ba ...a = 0, which we recall is enforced for n ≤ dλ a ...a = 0, and since λ a ...a vanishes at the identity, it must be identically zero. The remainingadjoint invariance conditions can be combined to imply that f abc λ abd = 0 , f af [ b | λ f | b b λ b b b ] = 0 , f a [ b b b λ b b b ] = 0 ,f db b c λ a a d − f da a [ c λ b b ] d − f de [ b λ b c ] d λ a a e − f de [ a λ a ] d [ b λ b c ] e = 0 . (3.10)These conditions are sufficient to ensure that frame algebra is obeyed.We also define the generalized frame field E A , which is a section of N , through (cid:104) E A , E B (cid:105) = η AB C E C , (3.11)where η AB C is an invariant tensor of the E n ( n ) . For E this tensor is related to the symmetricinvariant (see the appendix for details) such that the explicit form of E A has components E a = r a , E a ...a = 4 λ [ a a a r a ] + r a ...a , E a (cid:48) ,a ...a = ( λ a ...a r a (cid:48) − λ a (cid:48) [ a a λ a a a r a ] ) − λ a (cid:48) [ a a r a ...a ] + jr a (cid:48) r a ...a . (3.12)Here we denote r a ...a m = r a ∧ · · · ∧ r a m and make use of the j -wedge contraction of [32] todeal with mixed symmetry fields. One can consider now the action of the frame field E A onthese E A and by virtue of eq. (3.11), again find that they furnish the EDA algebra, albeit in adifferent representation as described in the appendix. The generalised frame field introduced above can be used as a compactification Ansatz withinExFT known as a generalised Scherk-Schwarz reduction. In this procedure all internal coor-dinate dependence is factorised into dressings given by the generalised frame. The algebra in For a p + 1-form α and a ( n − p )-form β , we define( jα ∧ β ) i,i ...i n = n ! p !( n − p )! α i [ i ...i p β i p +1 ...i n ] . (3.13) and representations of E as X ABC = d ABD Z CD + 10 d ADS d BRT d CDR Z ST − ϑ [ A δ CB ] − d ABD d CDE ϑ E . (3.14)The components of the antisymmetric Z AB are determined to be Z ab = 0 ,Z a a b = − · √ f ddbc ...c (cid:15) a a c ...c (= 0) ,Z a ...a b = − √ f ddbc (cid:15) a ...a c ,Z a a , b b = − √ ( f [ a c c c (cid:15) a ] c c c b b − f [ b c c c (cid:15) b ] c c c a a ) ,Z a a , b ...b = − √ ( f a a c (cid:15) cb ...b − f [ b b c (cid:15) b b b ] ca a ) ,Z a ...a , b ...b = 0 , (3.15)and those of ϑ A (sometimes called the trombone gauging) to be ϑ a = f acc , ϑ a a = − f cca a , ϑ a ...a = − f cca ...a . (3.16) ρ -twisting In the context of DFT, Yang-Baxter deformations can be understood as the O( d, d ) transfor-mation, generated by a bivector, acting on a Drinfel’d double with vanishing dual structureconstants [41–46]. The bivector that generates the transformation is then related to the classi-cal r -matrix, the dual structure constants are coboundaries and the requirement that the O( d, d )transformed algebra is a Drinfel’d double is precisely the classical Yang-Baxter equation.This suggests a natural generalisation of Yang-Baxter deformations to EDAs [13, 14]. Webegin with an EDA (cid:98) d with only the structure constants, f abc , corresponding to a maximally11sotropic Lie subalgebra g , non-vanishing and f abcd = f ab ...b = 0, i.e.ˆ T a ◦ ˆ T b = f abc ˆ T c , ˆ T a ◦ ˆ T b b = 2 f ac [ b ˆ T b ] c , ˆ T a ◦ ˆ T b ...b = − f ac [ b ˆ T b ...b ] c , ˆ T a a ◦ ˆ T b = 3 f [ c c [ a δ a ] b ] ˆ T c c , ˆ T a a ◦ ˆ T b b = f c c [ a ˆ T a ] b b c c , ˆ T a a ◦ ˆ T b ...b = 0 , ˆ T a ...a ◦ ˆ T b = 5 f bc [ a ˆ T a ...a ] c + 10 f c c [ a δ a b ˆ T a a a ] c c , ˆ T a ...a ◦ ˆ T b b = 0 , ˆ T a ...a ◦ ˆ T b ...b = 0 . (4.1)We denote the structure constants collectively as ˆ X ABC .We now perform an E transformation of the above EDA by a trivector, ρ abc , and hex-avector, ρ a ...a , which will play the analogue of the classical r -matrix. The corresponding E group element is given by C AB ≡ (cid:0) e ρ a ...a R a ...a e ρ a a a R a a a (cid:1) AB , (4.2)in which the generators R a a a and R a ...a are specified in the appendix. Explicitly we havethat ( C AB ) = δ ba ρ ba a √ δ a a b b ˜ ρ b ; a ··· a √
5! 20 δ [ a a b b ρ a a a √
2! 5! δ a ··· a b ··· b , (4.3)where ˜ ρ b ; a ...a ≡ ρ ba ...a + 5 ρ b [ a a ρ a a a ] . (4.4)Equivalently, we twist the generators by the group element (4.2) resulting in T a = ˆ T a , T a a = ˆ T a a + ρ ba a ˆ T b ,T a ...a = ˆ T a ...a + 10 ρ [ a a a ˆ T a a ] + ˜ ρ b ; a ...a ˆ T b . (4.5)For the twisted generators, we obtain T A ◦ T B = X ABC T C with X ABC ≡ C AD C BE ( C − ) F C ˆ X DEF . (4.6)We now require that the new algebra defines an EDA d . This imposes conditions on ρ abc and ρ a ...a and we will interpret these as analogues of the classical Yang-Baxter equation. From12he products T a ◦ T b b and T a ◦ T b ...b , the dual structure constants are identified as f ab b b = 3 f ac [ b ρ | c | b b ] , f ab ...b = 6 f ac [ b | ˜ ρ c ; | b ...b ] (= 0) , (4.7)which take the form of (twisted) coboundaries (2.30) and the (= 0) holds for d . The remainingproducts impose a number of conditions, of which the following is particularly intriguing3 f d d [ b ρ b b ] d ρ a a d = f d d [ a ˜ ρ a ]; b b b d d . (4.8)This is a natural generalisation of the classical Yang-Baxter (YB) equation which we willelaborate more on later. In general, we get a further set of conditions which are required toensure that the new algebra defines an EDA d . Mostly these additional conditions appearrather cumbersome but we note the requirement that ρ a a b f a a c = 0 . (4.9)With f ab ...b = 0, the Bianchi identity for f abc together with the generalised Yang-Baxterequation eq. (4.8) and compatibility condition eq. (4.9) imply the fundamental identity for f ab ...b . Indeed since the Leibniz identity (2.2) is E -invariant, it is guaranteed to hold for X ABC . Therefore, we see that the generalised Yang-Baxter equation (4.8) together with theother conditions obtained by imposing that the new algebra is an EDA imply that the newdual structure constants satisfy their fundamental identities (2.13) and (2.14) and the condition(2.17).In [21], a different approach was taken using a generalisation of the open/closed stringmap to propose a generalisation of the classical YB equation for a trivector deformation of11-dimensional supergravity. The approach of [21] is not limited to group manifolds, unlike thepresent case, but also only considers trivector deformations. However, when specialising [21] togroup manifolds and considering our deformations with ρ a ...a = 0, the resulting equation of [21]is different and, in particular, weaker than the YB equation we find here (4.8) with ρ a ...a = 0, orindeed the SL(5) case discussed in [13, 14]. Indeed, as shown in [22] based on explicit examples,the proposed YB-like equation of [21] is not sufficient to guarantee a solution of the equations ofmotion of 11-dimensional supergravity, while our deformations subject to the above conditionspreserve the equations of motion of 11-dimensional supergravity by construction. ρ -twisting The trivector ρ a a a and hexavector ρ a ...a define 3- and 6-brackets via (2.30). First define themaps ρ : g ∗ ∧ g ∗ −→ g , ˜ ρ : Λ g ∗ −→ g , (4.10)13s ρ ( x , x ) = ρ abc ( x ) b ( x ) c T a , ˜ ρ ( x , . . . , x ) = ˜ ρ a ; a ...a ( x ) a . . . ( x ) a T a , ∀ x , . . . , x ∈ g ∗ . (4.11)This allows the generalised Yang-Baxter equation eq. (4.8) to be cast in a basis independentway x (cid:0)(cid:0) ad ρ ( y ,y ) ρ (cid:1) ( x , x ) (cid:1) + y ( ˜ ρ (∆( y ) , x , x , x )) − y ( ˜ ρ (∆( y ) , x , x , x )) = 0 , (4.12)for all y , y , x , x , x ∈ g ∗ . Note that the first term in (4.12) is automatically antisymmetricin ( x , x , x ) due to antisymmetry of ρ a a a .Then, the associated 3- and 6-brackets are defined as { x , x , x } = ad ρ ( x ,x ) x + ad ρ ( x ,x ) x + ad ρ ( x ,x ) x , { x , . . . , x } = ad ˜ ρ ( x ,...,x ) x + cyclic permutations , (4.13)for all x , . . . , x ∈ g ∗ . Alternatively, to match more with the usual discussion of classical r -matrices in integrability, we can use the Cartan-Killing form on g to define 3- and 6-bracketson g . For this, it is more convenient to define ρ (cid:48) and ρ (cid:48) as ρ (cid:48) : g ∧ g −→ g , ˜ ρ (cid:48) : Λ g −→ g , (4.14)with ρ (cid:48) ( x , x ) = ρ ( κ ( x ) , κ ( x )) , ˜ ρ (cid:48) ( x , . . . , x ) = ˜ ρ ( κ ( x ) , . . . , κ ( x )) , (4.15)and where κ is the Cartan-Killing metric viewed as a map κ : g −→ g ∗ . Now, the 3- and6-brackets on g are defined as { x , x , x } = (cid:2) x , ρ (cid:48) ( x , x ) (cid:3) + (cid:2) x , ρ (cid:48) ( x , x ) (cid:3) + (cid:2) x , ρ (cid:48) ( x , x ) (cid:3) , { x , . . . , x } = (cid:2) x , ˜ ρ (cid:48) ( x , . . . , x ) (cid:3) + cyclic permutations . (4.16)The generalised Yang-Baxter equation (4.8) together with the other constraints requiredsuch that the new algebra is an EDA, such as (4.9), imply that the 3- and 6-brackets definedabove in (4.13) and (4.16) satisfy their fundamental identities (2.13) and (2.14). To understand better the generalised YB equation obtained above, let us adopt a tensor productnotation ρ = ρ abc T a ⊗ T b ⊗ ⊗ T c ⊗ g . Assuming that (4.9) holds and that ρ a ...a = 0 we have that (4.8)14ecomes [ ρ , ρ ] + [ ρ , ρ ] + [ ρ , ρ ]+ 12 (cid:0) [ ρ + ρ , ρ ] + [ ρ + ρ , ρ ] + [ ρ + ρ , ρ ] (cid:1) = 0 . (4.17)Introducing a (anti-)symmetrizer in the tensor product σ [123] , [45] , allows this equation to beconcisely given as σ [123] , [45] [ ρ + ρ , ρ ] = 0 . (4.18)Suppose that we have a preferred q ∈ g such that ρ = r ⊗ q + r ⊗ q − r ⊗ q , r = (cid:88) a,b (cid:54) = q r ab T a ⊗ T b = − r , (4.19)with r neutral (i.e. [ r , q ] = 0) then we find eq. (4.17) becomes YB [12 | | ⊗ q ⊗ q − YB [12 | | ⊗ q ⊗ q = 0 , (4.20)in which YB = [ r , r ] + [ r , r ] + [ r , r ] , (4.21)is the classical Yang-Baxter equation for r .Recall that the classical YB equation arises from the quantum one R R R = R R R (4.22)as the leading terms in the ‘classical’ expansion R = 1 + (cid:126) r + O ( (cid:126) ). An obvious questionis if there is an equivalent ‘quantum’ version of eq. (4.18)? We give here one proposal (with noclaim of first principle derivation or uniqueness) for such a starting point. Let us define in thesemi-classical limit R i ; jk = 1 + (cid:126) ρ ijk + O ( (cid:126) ) , (4.23) R ij ; kl = 1 + (cid:126) ρ ijk + ρ ijl − ρ ikl − ρ jkl ) + O ( (cid:126) ) . (4.24)Then eq. (4.17) follows from σ [123] , [45] R R R = σ [123] , [45] R R R . (4.25)This view point is very suggestive that this may just represent a standard Yang-Baxter equationfor the scattering of ∧ g , ∧ g and g obtained by S-matrix fusion. Here we leave an explorationof this as an open direction; further work is required to understand which quantum R-matricesgive rise under S-matrix fusion to an R i ; jk and R ij ; kl with the expansion (4.23) and what are We use lower case Roman indices to denote tensor product locations. igure 1: A proposed schematic for the generalised Yang-Baxter equation. The red lines indicateanti-symmetrisation and the black circle is a contact term that gives rise in the semi-classical limitto a contribution involving ρ . the resultant ρ ijk . Conversely one might ask if there exist solutions of (4.25) compatible (4.23)but that are not obtained from fusion?Restoring ρ a ...a we can amend this equation to σ [123] , [45] [ ρ + ρ , ρ ] = 12 ( ρ + ρ ) , (4.26)where, for example, ρ ≡ ρ abcdef T a ⊗ T b ⊗ T c ⊗ T d ⊗ [ T e , T f ] . (4.27)This is somewhat suggestive of a contact term in the YB relation that may lead to a quantumversion of the form σ [123] , [45] R R R − σ [123] , [45] R R R = σ [123] , [45] R . (4.28)Pictographically this is indicated in figure 1. In this section we wish to present a range of examples of the EDA, both of coboundary type andotherwise. We will give some broad general classes that correspond to embedding the algebraicstructure underlying existing T-dualities of the type II theory. In addition, in the absence of acomplete classification, here we provide a selection of specific examples.
When the subalgebra { T a } is Abelian, arbitrary ρ a a a and ρ a ...a are solutions of Yang–Baxter-like equations. However f ab b b = 0 and f ab ...b = 0 and the EDA is Abelian. The algebraic structure corresponding to non-Abelian T-duality is a semi-abelian Drinfel’ddouble i.e. a double constructed from some n − U (1) n − (or perhaps R n − ) factor.16n analogue here would be to take f abc (cid:54) = 0 and f ab ...b = 0, this however is not especiallyinteresting. More intriguing is to consider the analogue of the picture after non-Abelian T-dualisation has been performed in which the U (1) n − would be viewed as the physical space.This motivates the case of semi-Abelian EDAs with f abc = 0 but f ab ...b (cid:54) = 0.In this case the Leibniz identities reduce to the fundamental identities f adc c f db b b − f dc c [ b f ab b ] d = 0 , f ab ··· b = 0 . (5.1)Each solution for this identity gives an EDA. To identify these one can use existing classificationefforts and considerations of three algebras that followed in light of their usage [47] to describetheories of interacting multiple M2 branes.The first case to consider are the Euclidean three algebras, such that f b ...b = f ab ...b δ ab istotally antisymmetric. Here the fundamental identity is very restrictive and results in a uniquepossibility: the four-dimensional Euclidean three algebra [48–50], whose structure constants arejust the antisymmetric symbol, complemented with two U (1) directions. Relaxing the require-ment of a positive definite invariant inner product allows a wider variety [51–56]. Dispensing therequirement of an invariant inner product (which thus far appears unimportant for the EDA)allows non-metric three algebras [57–59]. r -matrix EDAs We now consider coboundary EDAs given in terms of an r -matrix as in eq.(4.19) obeying theYB equation (4.21). Splitting the generators of g into T ¯ a with ¯ a = 1 , . . . , T (identifiedwith the generator q appearing in (4.19)) we have the non-vanishing components ρ ¯ a ¯ b = r ¯ a ¯ b .Furthermore the condition (4.9) requires that r ¯ a ¯ b f ¯ a ¯ b = r ¯ a ¯ b f ¯ a ¯ b ¯ c = r ¯ a ¯ b f ¯ a c = r ¯ a ¯ b f ¯ a = 0 , (5.2)in which the last two equalities match the statement that r is neutral under T . In such a setup,the dual structure constants are specified as f ¯ a ¯ b ¯ b = 2 f ¯ a ¯ c [¯ b r | ¯ c | ¯ b ] + f ¯ a r ¯ b ¯ b , f ¯ a ¯ b ¯ b ¯ b = 3 f ¯ a b r ¯ b ¯ b ] , f b ¯ b = f b ¯ b ¯ b = 0 . (5.3)Assuming further that ¯ g = span( T ¯ a ) is a sub-algebra of g then r ¯ a ¯ b defines an r -matrix on ¯ g obeying the YB equation. Consequently ˜ f ¯ b ¯ b ¯ a = − f ¯ a ¯ c [¯ b r | ¯ c | ¯ b ] are the structure constants of adual Lie algebra ¯ g R and ¯ d = ¯ g ⊕ ¯ g R is a Drinfel’d double. Thus we have a family of embeddingsof the Drinfel’d double into the EDA specified by f ¯ a and f ¯ a b . When g = ¯ g ⊕ u (1) is a directsum (such that f ¯ a = f ¯ a b = 0), then this is precisely an example of the non-metric threealgebra of [57–59]. We emphasise though that not every (coboundary) double can be embeddedin this way; one must still ensure that equation (5.2) holds. In addition there are three algebra structures [60,61] in which f dabc is not totally antisymmetric in its upperindices. These can be used to describe interacting 3d theories with lower supersymmetry. It is a unclear if theycould play role in the context of EDAs. .4 Explicit examples We now present a selection of explicit examples that illustrate coboundary and non-coboundaryEDAs. SO ( p, q )To illustrate that the EDA requirements are indeed quite restrictive we can first consider thecase of g = so ( p, q ) with p + q = 4. A direct consideration of the Leibniz identities reveals thatthere is no non-zero solution for f ab ...b (in fact the cocycle conditions alone determine this).Equally the Leibniz identities admit only trivial solutions in the case of iso ( p, q ) = so ( p, q ) (cid:110)R p + q + with p + q = 3. We consider an indecomposable nilpotent Lie algebra N , of [62] specified by the structureconstants f = 1 , f = 1 , f = c , f = 1 , f = 1 , f = c . (5.4)We find a family of solutions f = d , f = d , f = d , f = − d ,f = d , f = d , f = − d + d , (5.5)which indeed satisfies the closure constraints. In particular, if we choose c = 0, we can clearlysee that this EDA contains a 10D Drinfel’d double { T ¯ a , T ¯ a } (¯ a = 1 , . . . ,
5) as a Lie subalgebra.We can also find ρ abc by considering the coboundary Ansatz. Supposing c (cid:54) = 0, the generalsolution to the generalised Yang-Baxter equation and compatibility condition is ρ = e , ρ = e , ρ = e , ρ = − e , ρ = e , ρ = e , (5.6)where e = 0 or e = 0 . The corresponding structure constants are f = e , f = e , f = e , f = − e ,f = − e , f = e , f = e − e . (5.7)This means that only when d i ( i = 1 , . . . ,
5) have the form d = e , d = e , d = e , d = − e , d = e , (5.8)and satisfy d = 0 or d = 0, the cocycle becomes the coboundary. Here we introduced a parameter c for convenience, which is 1 in [62]. .4.3 An example with ρ In the previous example ρ is absent. By considering the Lie algebra of the form g = g ⊕ u (1) ⊕ u (1), where g denotes a real 4D Lie algebra that is classified in [63], one can constructa number of examples (based on unimodular Lie algebras) that admit ρ . To illustrate thislet us consider the case that g = A , specified by structure constants f = 1 , f = 1 . (5.9)We find the generalised Yang-Baxter and compatibility equations admit the following family ofsolutions: ρ = d , ρ = d , ρ = d , ρ = d d d d , ρ = d d d d , ρ = d ,ρ = d , ρ = d , ρ = d , ρ = d , ρ = 2 d d , ρ = d . (5.10)The corresponding dual structure constants are f = 2 d d , f = d , f = d , f = 2 d d ,f = − d d d d , f = − d d d d , f = − d , f = − d . (5.11) r -matrix EDA In order to find a non-trivial example of the r -matrix EDAs, we consider a solvable Lie algebra N αβ , of [64] defined by structure constants: f = 1 , f = α , f = 1 , f = 1 , f = 1 , f = β , f = − , (5.12)where α + β (cid:54) = 0 . This algebra contains a subalgebra generated by ¯ g = span( T ¯ a ) (¯ a = 1 , . . . , f ¯ a b (cid:54) = 0 and f ¯ a (cid:54) = 0.Supposing αβ (cid:54) = 0, we find the general solution for ρ abc is given by ρ = c , ρ = − c , ρ = c , ρ = c , (5.13)where c = 0 when α (cid:54) = β . The corresponding dual structure constants are f = (1 + α ) c , f = − (1 + α ) c , f = (1 + α ) c , f = (2 + α ) c ,f = (1 + β ) c , f = − (1 + β ) c , f = (2 + β ) c , f = (1 + β ) c ,f = − c , f = − c . (5.14)This solution contains an r -matrix EDA as a particular case c = c = 0 . In the notation of [63] examples with ρ (cid:54) = 0 are found when g is one of the following: A , + u (1), A − , + u (1), A , + u (1), A , , A − , , A a,b, − ( a + b )4 , , A − b,b , Conclusion and Outlook
In this work we have consolidated the exploration of exceptional Drinfel’d algebras introducedin [13,14] extending the construction to the context of the E exceptional group. The algebraicconstruction here requires the introduction of a new feature: we have to consider not only aLie algebra g together with a three-algebra specified by f ≡ f ab ...b as in [13, 14], but we haveto also include a six-algebra f ≡ f ab ...b . The Leibniz identities that the EDA must obeyenforce a set of fundamental (Jacobi-like) identities for the three- and six-algebra as well assome compatibility conditions. These compatibility conditions require that f be a g -cocycleand f be an f -twisted g -cocycle. In terms of the g coboundary operator d this can be statedas df = 0 , df + f ∧ f = 0 . (6.1)We can solve this requirement with a coboundary Ansatz, f = dρ and f = dρ + ρ ∧ dρ ,reminiscent of the way a Drinfel’d double can be constructed through an r -matrix. Indeed, wefind a generalised version for the Yang Baxter equation for ρ , concisely expressed as σ [123] , [45] [ ρ + ρ , ρ ] = 12 ( ρ + ρ ) . (6.2)We proposed a ‘quantum’ relation from which this classical equation can be obtained. Thisfeature, and the resultant interplay between one-, three-, and six-algebras, opens up manyinteresting avenues for further exploration.The construction of the EDA is closely motivated by considerations within exceptionalgeneralised geometry. We have shown how the EDA can be realised as a generalised Leibnizparallelisation of the exceptional generalised tangent bundle of a group manifold G . The datarequired to construct this mean that G is equipped with a 3-bracket and a 6-bracket whichinvites the consideration of Nambu-Lie groups.Now we come to solving the various constraint equations that govern the structure of theEDA. The first thing to note is that due to the dimension, the only solutions to the fundamentalidentities have vanishing f (and consequently a trivial 6-bracket on G ). We believe howeverthat in higher dimension this condition is less stringent and that there will solutions for whichthe structure described above is exhibited in full.We then provided a range of examples that illustrate the various features here. We have ex-amples with and without Drinfel’d double subalgebras, and examples that are both of cobound-ary type (specified by a ρ and ρ ) and not of coboundary type. All of the coboundary examplespresented here (and indeed in all the numerous other examples we have found) can be obtainedfrom the procedure of ρ -twisting i.e. starting with a semi-Abelian EDA and applying an E transformation parametrised by the ρ and ρ . Despite the dimensionality induced restriction to f = 0, there are examples for which ρ (cid:54) = 0. We provide examples where ρ can be parametrisedin terms of a Yang-Baxter r -matrix for a lower dimensional algebra, as well as where this is notthe case.There are several exciting open directions here that we share in the hope that others may20ish to develop them further: • Extensions of the EDA to E and higher are likely to shed further light on the structuresinvolved. As the space gets larger there is more scope to find interesting solutions. • It would be interesting to develop a more general classification of EDA solutions. • One feature of the EDA is that they may admit multiple decompositions into physicalspaces, and a resultant notion of duality. Further development should go into this veryinteresting aspect. • Here we make some robust requirements that result in structures compatible with maxi-mally supersymmetric gauged supergravities. It would likely be interesting to see how therequirements of the EDA can be consistently relaxed to lower supersymmetric settings,for example using [65]. • On a mathematical note perhaps the most intriguing area of all is to develop the ‘quantum’equivalent of the classical EDA proposed here.
EM is supported by ERC Advanced Grant “Exceptional Quantum Gravity” (Grant No.740209).YS is supported by JSPS Grant-in-Aids for Scientific Research (C) 18K13540 and (B) 18H01214.DCT is supported by The Royal Society through a University Research Fellowship
Gener-alised Dualities in String Theory and Holography
URF 150185 and in part by STFC grantST/P00055X/1; FWO-Vlaanderen through the project G006119N; and by the Vrije Univer-siteit Brussel through the Strategic Research Program “High-Energy Physics”. DCT thanksChris Blair collaboration on related work and Tim Hollowood for informative communications. A E × R + Algebra and conventions
The matrix representations of the E n ( n ) generators ( t α ) AB in the R -representation are K cd ≡ δ ca δ bd − δ a a [ b | d | δ cb ]
00 0 − δ a ...a [ b ...b | d | δ cb ] + δ cd − n ,R c c c ≡ − δ c c c ab b √
00 0 − δ a a c c c b ...b √
5! 2! , R c ...c ≡ − δ c ...c ab ...b √ ,R c c c ≡ δ ba a c c c √ δ a ...a b b c c c √
2! 5! , R c ...c ≡ δ ba ...a c ...c √ , (A.1)21nd that of the R + generator is simply ( t ) AB = − δ BA .The non-vanishing components of symmetric invariant tensor of E , d ABC , are d ab b c ...c = 2! δ a [ b (cid:15) b ] c ...c √ , d a a b b c c = (cid:15) a a b b c c √ . (A.2)and those of d ABC are given by the same with indices in opposite positions. Using this we havethe useful tensor given by η AB C ≡ √ d ABD k D C , (A.3)in which, k A B connects R = and R = ,( k A B ) ≡ δ ab (cid:15) b ··· b √ − (cid:15) a a b ··· b √
2! 4! (cid:15) a ··· a b √ , ( k A B ) ≡ (cid:15) b ··· b a √ − (cid:15) b b a ··· a √
2! 4! δ ab (cid:15) a ··· a √ . (A.4)This plays a similar role to that of the O ( d, d ) invariant inner product of generalised geom-etry and indeed can be used to construct the Y-tensor [36] given by Y ABCD = η AB E η CD E =10 d ABE d CDE that is ubiquitous in exceptional field theory/generalised geometry.The invariance of η AB C under E n ( n ) × R + symmetry can be expressed as( t ˆ α ) AD η DB C + ( t ˆ α ) BD η AD C = η AB D ( t ˆ α ) DC . (A.5)From this relation, we find the matrix representation ( t α ) AB in the R -representation is K cd ≡ − δ ad δ cb − δ a ...a de e e δ ce e e b ...b
00 0 − δ a ...a de ...e δ ce ...e b ...b δ a (cid:48) b (cid:48) − δ a ...a b ...b δ a (cid:48) d δ cb (cid:48) + 2 δ cd − n ,R c c c ≡ −√ δ ac c c b ...b
00 0 − √ δ a ...a d d b ...b δ c c c b (cid:48) d d , R c ...c ≡ √ δ c ...c b ...b δ a (cid:48) b (cid:48) ,R c c c ≡ √ δ a ...a bc c c √ δ a ...a b ...b d d δ a (cid:48) d d c c c , R c ...c ≡ −√ δ a ...a c ...c δ a (cid:48) b (cid:48) . (A.6)In terms of these generators we can express the EDA product as, T A ◦ T B = (cid:98) Θ A ˆ α ( t ˆ α ) BC T B , (A.7)where { t ˆ α } ≡ { t , t α } , { t α } ≡ { K ab , R a a a , R a ··· a , R a a a , R a ··· a } . The explicit form of22 Θ A ˆ α is as follows: (cid:98) Θ aα t α ≡ f abc K bc + f ac c c R c c c + f ac ··· c R c ··· c , (cid:98) Θ a a α t α ≡ − f c c [ a R a ] c c − f cda a K cd , (cid:98) Θ a ··· a α t α ≡ − f c c [ a R a ··· a ] c c + 10 f c [ a a a R a a ] c − f cda ··· a K cd , (cid:98) Θ a ≡ f acc − n , (cid:98) Θ a a ≡ − f cca a − n , (cid:98) Θ a ··· a ≡ − f cca ··· a − n . (A.8)We can now recast the algebra of frame fields (3.8) as L E A E B = − (cid:98) Θ A ˆ α ( t ˆ α ) BC E C , (A.9)such that making us of (3.7), (3.11), and (A.5), we can easily find that the generalized framefield in the R -representation also transforms covariantly as L E A E B = − (cid:98) Θ A ˆ α ( t ˆ α ) BC E C . (A.10) References [1] C. Hull and P. Townsend,
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