Poisson-Lie transformations and Generalized Supergravity Equations
aa r X i v : . [ h e p - t h ] J a n Poisson–Lie transformations and GeneralizedSupergravity Equations
Ladislav Hlavatý ∗ Faculty of Nuclear Sciences and Physical Engineering,Czech Technical University in Prague,Czech Republic
Ivo Petr † Faculty of Information Technology,Czech Technical University in Prague,Czech Republic
January 29, 2021
Abstract
In this paper we investigate Poisson–Lie transformation of dilatonand vector field J appearing in Generalized Supergravity Equations.While the formulas appearing in literature work well for isometricsigma models, we present examples for which Generalized Supergrav-ity Equations are not preserved. Therefore, we suggest modificationof these formulas. Contents ∗ hlavaty@fjfi.cvut.cz † ivo.petr@fit.cvut.cz Poisson–Lie transformations and Generalized SupergravityEquations 64 Examples 9 (1 | . . . . . . . . 94.1.1 Identity (1 | → (1 | and full duality (1 | → (5 | . . 104.1.2 Plurality (1 | → (6 − | ii . . . . . . . . . . . . . . . . 114.2 Poisson–Lie plurality on Drinfel’d double (1 | . . . . . . . . 124.2.1 Identity (1 | → (1 | . . . . . . . . . . . . . . . . . . . 134.2.2 Full duality (1 | → (4 | . . . . . . . . . . . . . . . . . 134.2.3 Plurality (1 | → (6 − | ii . . . . . . . . . . . . . . . . 144.2.4 Plurality (1 | → ( ii | − ) . . . . . . . . . . . . . . . . 15 Formula for Poisson–Lie transformation [1] of dilaton field accompanyingsigma model background was given long ago in [2]. Its limitations discussedin [3] concern the problem of possible appearance of unwanted ”dual” coor-dinates of Drinfel’d double in the transformed dilaton. The problem waspartially solved in [4, 5] for special cases where transformations of relevantcoordinates of Drinfel’d double are linear. The price we had to pay was thatthat in order to keep invariance with respect to Poisson–Lie transformationswe had to replace the beta function equations by Generalized SupergravityEquations [6] containing not only dilaton, but also additional vector fieldsfor which correct transformation formulas need to be found. Unfortunately,there are cases for which the transformation of relevant coordinates is notlinear or the Poisson–Lie formulas do not provide solutions of GeneralizedSupergravity Equation.In the following we have chosen several examples for which the problemof unwanted coordinates in the transformed dilatons does not appear, andstill, it turns out that the original formula [5, 7, 8, 9] for Killing vector field J , which works well for isometric initial sigma models, fails. These arethe cases where the initial sigma models are constructed from Manin triple ( d , g , ˜ g ) where ˜ g is neither Abelian nor unimodular. The purpose of this noteis to extend the validity of Poisson–Lie formulas to these cases. Beside that2or NS-NS Generalized Supergravity Equations it is not necessary to haveformulas for both dilaton and Killing vector J and one only needs formulafor Poisson–Lie transformation of X-form that combines these two. It is givenas well. Here we shall recapitulate well known basics of Poisson–Lie T-plurality withspectators [1, 2, 10] to establish notation.Sigma models in curved background are given by Lagrangian L = ∂ − φ µ F µν ( φ ) ∂ + φ ν , φ µ = φ µ ( σ + , σ − ) , µ = 1 , . . . , n + d (1)where tensor field F = G + B defines metric and torsion potential (Kalb–Ramond field) of the target manifold M .Assume that there is d -dimensional Lie group G with free action on M .The action of G is transitive on its orbits, hence we may locally consider M ≈ ( M / G ) × G = N × G , and introduce adapted coordinates { x µ } = { s α , x a } , α = 1 , . . . , n = dim N , a = 1 , . . . , d = dim G (2)where x a are group coordinates and s α label the orbits of G . s α are treatedas spectators in Poisson–Lie transformations.Poisson–Lie duality/plurality is based on the possibility to pass betweenvarious decompositions of Lie algebra of Drinfel’d double D = ( G | e G ) intoManin triples ( d , g , ˜ g ) . Poisson–Lie dualizable sigma models on N × G aregiven by tensor field F of the form F ( s, x ) = E ( x ) · ( n + d + E ( s ) · Π( x )) − · E ( s ) · E T ( x ) . (3) E ( s ) is spectator-dependent ( n + d ) × ( n + d ) matrix, Π( x ) is given by sub-matrices a ( x ) and b ( x ) of the adjoint representation as ad g − ( e T ) = b ( x ) · T + a − ( x ) · e T , Π( x ) = (cid:18) n b ( x ) · a − ( x ) (cid:19) , and matrix E ( x ) reads E ( x ) = (cid:18) n e ( x ) (cid:19) e ( x ) is d × d matrix of components of right-invariant Maurer–Cartanform ( dg ) g − on G .For many Drinfel’d doubles several decompositions may exist. Supposethat we have sigma model on N × G and the Drinfel’d double splits intoanother pair of subgroups b G and ¯ G . Then we can apply the full frameworkof Poisson–Lie T-plurality [1, 2] and find background for sigma model on N × b G .Let Manin triples ( d , g , ˜ g ) and ( d , ˆ g , ¯ g ) be two decompositions of d intodouble cross sum of subalgebras that are maximally isotropic with respect tonon-degenerate symmetric bilinear form h ., . i on the Lie algebra of Drinfel’ddouble. Pairs of mutually dual bases T a ∈ g , e T a ∈ ˜ g and b T a ∈ ˆ g , ¯ T a ∈ ˆ g , a = 1 , . . . , d, then must be related by transformation (cid:18) b T ¯ T (cid:19) = C · (cid:18) T e T (cid:19) (4)where C is an invertible d × d matrix. For the following formulas it will beconvenient to introduce d × d matrices P, Q, R, S as (cid:18) T e T (cid:19) = C − · (cid:18) b T ¯ T (cid:19) = (cid:18) P QR S (cid:19) · (cid:18) b T ¯ T (cid:19) (5)and extend these to ( n + d ) × ( n + d ) matrices P = (cid:18) n P (cid:19) , Q = (cid:18) n Q (cid:19) , R = (cid:18) n R (cid:19) , S = (cid:18) n S (cid:19) to accommodate the spectator fields.The sigma model on N × b G obtained from (3) via Poisson–Lie T-pluralityis given by tensor field b F ( s, ˆ x ) = b E (ˆ x ) · b E ( s, ˆ x ) · b E T (ˆ x ) , b E (ˆ x ) = (cid:18) n b e (ˆ x ) (cid:19) , (6)where b e (ˆ x ) is d × d matrix of components of right-invariant Maurer–Cartanform ( d ˆ g )ˆ g − on b G and b E ( s, ˆ x ) = (cid:16) n + d + b E ( s ) · b Π(ˆ x ) (cid:17) − · b E ( s ) = (cid:16) b E − ( s ) + b Π(ˆ x ) (cid:17) − . (7)4he matrix b E ( s ) is obtained from E ( s ) in (3) by formula b E ( s ) = ( P + E ( s ) · R ) − · ( Q + E ( s ) · S ) , (8)and b Π(ˆ x ) = (cid:18) n b b (ˆ x ) · b a − (ˆ x ) (cid:19) ,ad ˆ g − ( ¯ T ) = b b (ˆ x ) · b T + b a − (ˆ x ) · ¯ T .
Conformal invariance up to the first loop requires introduction of dilatonfield Φ satisfying beta function equations R µν − H µρσ H ρσν + 2 ∇ µ ∇ ν Φ , (9) − ∇ ρ H ρµν + ∇ ρ Φ H ρµν , (10) R − H ρστ H ρστ + 4 ∇ µ ∇ µ Φ − ∇ µ Φ ∇ µ Φ (11)where H ρµν = 12 ( ∂ ρ B µν + ∂ µ B νρ + ∂ ν B ρµ ) , and ∇ µ are covariant derivatives with respect to metric G .Formula for transformation of dilaton under Poisson–Lie T-plurality wasgiven in [2] as b Φ(ˆ x ) = Φ( y ) + 12 L ( y ) − b L (ˆ x ) (12)where y represent coordinates of group G , Φ( y ) is the dilaton of the initialmodel, and terms L ( y ) , b L (ˆ x ) read L ( y ) = ln (cid:12)(cid:12)(cid:12) det [( + Π( y ) E ( s )) a ( y )] (cid:12)(cid:12)(cid:12) , b L (ˆ x ) = ln (cid:12)(cid:12)(cid:12) det h(cid:16) + b Π(ˆ x ) b E ( s ) (cid:17) N − b a (ˆ x ) i (cid:12)(cid:12)(cid:12) , with N = P T − R T E ( s ) . exp( − (0) ( y )) := exp( − y )) (det G ( y )) / det u ( y )= exp( − b Φ (0) (ˆ x )) := exp( − b Φ(ˆ x )) (det b G (ˆ x )) / det ˆ u (ˆ x ) (13)where G and b G are metrics of sigma models on G resp. b G and u , ˆ u arecorresponding matrices of components of left-invariant forms. In the case that the initial dilaton Φ( y ) depends on coordinates y k , we have toexpress these in terms of ˆ x, ¯ x to get explicit form of transformed dilaton. Thiscan be done using relation between two different decompositions of Drinfel’ddouble elements g ( y )˜ h (˜ y ) = b g (ˆ x )¯ h (¯ x ) , g ∈ G , ˜ h ∈ ˜ G , b g ∈ b G , ¯ h ∈ ¯ G . (14)The origin of the puzzle discussed in [3] is that if ∂y k ∂ ¯ x j = 0 , formulas (12), (13) give b Φ that may depend not only on coordinates ˆ x of thegroup b G but also on coordinates ¯ x of ¯ G .Partial solution of this problem was given in [5] for the case of lineardependence y k (ˆ x, ¯ x ) = ˆ d kj ˆ x j + ¯ d kj ¯ x j . (15)It was suggested that in this case we can set y k = ˆ d km ˆ x m in the formula (12)and extend the beta function equations to (NS-NS) Generalized Supergravity6quations [6, 11] R µν − H µρσ H ρσν + ∇ µ X ν + ∇ ν X µ , (16) − ∇ ρ H ρµν + X ρ H ρµν + ∇ µ X ν − ∇ ν X µ , (17) R − H ρστ H ρστ + 4 ∇ µ X µ − X µ X µ (18)where X µ = ∂ µ Φ + J ν F νµ , (19) Φ is the dilaton and J is a vector field. For vanishing J the usual betafunction equations are recovered.Authors of [5, 9] give formula allowing to find components of b J forPoisson–Lie transformed sigma model as b J α = 0 , α = 1 , . . . , n = dim N , b J dim N + m (ˆ x ) = (cid:18)
12 ¯ f abb − ∂ Φ ( y ) ∂y k (cid:12)(cid:12)(cid:12) y = b D · ˆ x ¯ d ka (cid:19) b V am (ˆ x ) , (20)where a, b, k, m = 1 , . . . , dim G , b D = (cid:18) n b d (cid:19) , (21) b V a are left-invariant fields of the group b G , and ¯ f bac are structure constantsof the Lie algebra of ¯ G .From the form of equations (16)–(18) one can see that only one-form X is important for their satisfaction, not separately dilaton Φ and vector field J . Therefore, for Poisson–Lie transformation of Generalized SupergravityEquations it would be sufficient to know only Poisson–Lie transformation ofthe one-form X and of the tensor F .Note that Φ and J are not defined uniquely as the form X is invariantwith respect to gauge transformation Φ( x ) Φ( x ) + λ ( x ) , J κ
7→ J κ − ∂ ν λ F νκ . (22)This means that we can always choose dilaton vanishing. On the other hand,if X is closed, we can choose X = d Φ and J vanishing in which case General-ized Supergravity Equations of Motion become usual beta function equations.7oreover, note that even the form X satisfying equations (16)–(18) is notunique. Namely, if X µ satisfy the Generalized Supergravity Equations, then X ′ µ := X µ + χ µ , (23)where ∇ ν χ µ = 0 , ( X µ + 2 χ µ ) χ µ = 0 , (24)satisfy the equations as well. Simple example that was mentioned in [13] issigma model given by flat Minkowski metric and X µ = (0 , , , , χ µ = (cid:18) , , , − x (cid:19) . In the following we verify whether backgrounds obtained by Poisson–Lie T-plurality supported by b Φ and b J obtained from (12) and (20) satisfyGeneralized Supergravity Equations and present examples where this is nottrue. In all our examples ∂y k /∂ ¯ x j = 0 so it is not necessary to solve theequation (14). It turns out that formula (20) works well for isometric initialsigma models, but not always as we will show in the following examples.Therefore, it is desirable to modify the prescription for vector fields b J . Itseems that appropriate formula is b J dim N + m (ˆ x ) = 12 e f abb (cid:18) ∂ e y a ∂ ¯ x k b V km (ˆ x ) − ∂ e y k ∂ b x a b F km (cid:19) + (cid:18)
12 ¯ f abb − ∂ Φ ( y ) ∂y k (cid:12)(cid:12)(cid:12) y = b D · ˆ x ∂y k ∂ ¯ x a (cid:19) b V am (ˆ x ) (25)where b V a are left-invariant fields of the group b G , e f bac and ¯ f bac are structureconstants of Lie algebras of e G , ¯ G and b D is given by (21). This modificationdoes not change results of [5] and [9] because those papers deal with groups forwhich e f abb = 0 . Finally let us mention that b J obtained from (25) is not alwaysKilling of b F (which is not necessary for satisfaction of the NS-NS GeneralizedSupergravity Equations) but we can use the gauge transformation (23) toacquire this property.Having formulas (12) and (25) for dilatons and vector fields b J it is easyto write down formula for Poisson–Lie transformation of the form X b X µ (ˆ x ) = ∂ Φ ( y ) ∂y ν (cid:12)(cid:12)(cid:12) y = b D · ˆ x ∂y ν ∂ ˆ x µ − ∂ b L (ˆ x ) ∂ ˆ x µ + b J ν (ˆ x ) b F νµ (ˆ x ) (26)8here Φ ( y ) = Φ( y ) + L ( y ) , and b J ν (ˆ x ) are given by (25). Advantage of theformula (26) is that b X , differently from b Φ and b J , is invariant with respectto the gauge transformation (22). In this paper the groups G will be non-semisimple Bianchi groups. Theirelements will be parametrized as g = e x T e x T e x T where e x T e x T and e x T parametrize their normal subgroups. We will deal with backgrounds onfour-dimensional manifolds, hence dim N = 1 and we denote the spectator s as t . (1 | We shall start our discussion with tensor field F ( t, y ) = − t t + y + y y t + y + y y t + y + y − y t + y + y t + y t ( t + y + y ) − y y t ( t + y + y )0 − y t + y + y − y y t ( t + y + y ) t + y t ( t + y + y ) (27)specifying sigma model on Abelian group G with corresponding Drinfel’ddouble D = ( G | e G ) = (1 | whose non-trivial commutation relations read [ e T , e T ] = e T , [ e T , e T ] = e T , [ e T , T ] = − T , (28) [ e T , T ] = − T , [ e T , T ] = T , [ e T , T ] = T . Background (27), dilaton Φ( t, y ) = −
12 ln (cid:0) − t (cid:0) t + y + y (cid:1)(cid:1) (29)and Killing vector J = 2 ∂ ˆ x satisfy Generalized Supergravity Equations.Corresponding X-form with components X µ ( t, y ) = 1 t + y + y (cid:18) − t + y + y t , t , y , y (cid:19) (30) For typographic reasons we write coordinate indices as subscripts. To denote Drinfel’d doubles and Manin triples we use notation of [12], where slots in ( . | . ) denote numbers corresponding to three dimensional algebras in Bianchi classification.
9s not closed and we cannot get rid of the vector J by gauge transforma-tions, so reduction of Generalized Supergravity Equations to beta functionequations is not possible.Background (27) is actually non-Abelian dual of flat background b F ( t, ˆ x ) = − t e x t
00 0 0 e x t (31)studied frequently in the literature [14, 15, 16]. b F is invariant with respectto action of Bianchi 5 group and dilaton (29) and J were obtained via (12)and (20).Let us note that Φ ( t, y ) = − ln t and it is not necessary to solve equa-tion (14) for y to get transformed dilatons b Φ(ˆ x ) . Similar results can beobtained starting from Drinfel’d double (1 | . (1 | → (1 | and full duality (1 | → (5 | Let us check formulas (12) and (20) applying Poisson–Lie transformationwith C equal to identity matrix to (27) and (29). We recover the originalbackground and dilaton, but vector field b J = ∂ ˆ x obtained from (20) isdifferent from the initial one and Generalized Supergravity Equations arenot satisfied even in this simple case. Using (25) instead of (20) we getback Killing vector b J = 2 ∂ ˆ x and Generalized Supergravity Equations aresatisfied.By full duality given by C = D := (cid:18) d d (cid:19) (32)we get flat background (31), but dilaton b Φ( t, ˆ x ) = 12 ln (cid:0) e x (cid:1) (33)and vanishing vector b J obtained from formulas (12) and (20) do not satisfyGeneralized Supergravity Equations. 10n the other hand, equations (16)–(18) are satisfied for (31), (33) and b J ( t, ˆ x ) = − t ∂ ˆ x (34)that follows from (25). Corresponding X-form b X vanishes, and by gaugetransformation it is possible to eliminate b J while changing dilaton to b Φ ′ ( t, ˆ x ) = − c t e − ˆ x + c (35)with c , c arbitrary constants. Dilaton (35) and flat background (31) satisfybeta function equations. (1 | → (6 − | ii By Poisson–Lie plurality given by C (1 | → (6 − | ii = − − (36)we get background tensor b F ( t, ˆ x ) = − t e x ( ˆ x +1 ) t + e x ( ˆ x +1 ) t t + e x ( ˆ x +1 ) t e x ˆ x t + e x ( ˆ x +1 )0 − t t + e x ( ˆ x +1 ) t t + e x ( ˆ x +1 ) e x ˆ x t + e x ( ˆ x +1 )0 t e x ˆ x t + e x ( ˆ x +1 ) − e x ˆ x t + e x ( ˆ x +1 ) e x ( t + e x ) t e x ( ˆ x +1 ) + t (37)and dilaton b Φ( t, ˆ x ) = −
12 ln (cid:0) t e − x ∆ (cid:1) , ∆ = t + e x (cid:0) ˆ x + 1 (cid:1) . (38)Generalized Supergravity Equations are satisfied for b J ( t, ˆ x ) = 1 t ∂ ˆ x (39)11alculated via (25). Vector field b J = ∂ ˆ x obtained from formula (20) doesnot satisfy Generalized Supergravity Equations.Correct X-form with components b X µ ( t, ˆ x ) = 1∆ (cid:18) − t + e x (ˆ x + 1) t , e x (cid:0) ˆ x + 1 (cid:1) , t , e x ˆ x (cid:19) (40)is not closed and Generalized Supergravity Equations cannot be reducedto beta function equations. Beside that, vector field (39) is not Killing of(37). However, using the gauge transformation (23) with λ = ˆ x we get b Φ ′ ( t, ˆ x ) = b Φ( t, ˆ x ) + ˆ x and b J ′ ( t, ˆ x ) = 2 ∂ ˆ x that is Killing vector field of (37). X-form remains unchanged, of course. (1 | Next we shall investigate plural sigma models on Drinfel’d double (1 | withcommutation relations [ e T , e T ] = − e T + e T , [ e T , e T ] = − e T , [ e T , T ] = T , (41) [ e T , T ] = − T + T , [ e T , T ] = − T , [ e T , T ] = T , [ e T , T ] = − T . Background F ( t, y ) = − y y − y y − y +1 y − y y +1 ( y − y ) y − (42)on Abelian group G was obtained as non-abelian T-dual of flat background b F ( t, ˆ x ) = e − ˆ x ˆ x e − ˆ x e − ˆ x ˆ x e − x e − ˆ x (43)that is invariant with respect to the action of Bianchi 4 group.Background (42), dilaton Φ( t, y ) = −
12 ln (cid:0) − y (cid:1) (44)and Killing vector J = − ∂ ˆ x satisfy Generalized Supergravity Equations.Since Φ ( y ) = 0 , we once again do not need to solve (14).12 .2.1 Identity (1 | → (1 | To check formulas (20) and (25) we apply Poisson–Lie transformation with C equal to identity matrix to (42) and (44). We get the original background anddilaton. Formula (20) gives vector − ∂ ˆ x , while from (25) we obtain Killingvector b J = − ∂ ˆ x . For the former one Generalized Supergravity Equationsare not satisfied, for the latter they hold.Corresponding X-form b X ( t, ˆ x ) = ˆ x + 2ˆ x − d ˆ x (45)is closed and we can pass to dilaton b Φ ′ ( t, ˆ x ) = 32 ln(1 − ˆ x ) −
12 ln(ˆ x + 1) (46)that together with (42) satisfies beta function equations. (1 | → (4 | By full duality (1 | → (4 | we get flat background (43), but non-trivialdilaton b Φ( t, ˆ x ) = 12 ln (cid:0) e − x (cid:1) (47)obtained from (12) and vanishing vector field b J obtained from (20) do notsatisfy Generalized Supergravity Equations. Correct vector field for whichthese equations are satisfied is b J ( t, ˆ x ) = e ˆ x ∂ ˆ x (48)and follows from (25). Corresponding X-form vanishes and using gauge trans-formation (22) we can get b J ′ = 0 and dilaton b Φ ′ = 0 satisfying beta functionequations. 13 .2.3 Plurality (1 | → (6 − | ii Changing the decomposition of Drinfel’d double to (6 − | ii using matrix C (1 | → (6 − | ii = −
20 1 0 0 0 01 0 0 0 0 −
20 0 − − (49)we get background b F ( t, ˆ x ) = − e x (ˆ x +2ˆ x ) e x − − e ˆ x − − e x (ˆ x +2ˆ x ) e ˆ x − − e ˆ x +2 e x (ˆ x +2ˆ x ) e ˆ x +2 e x (50)and dilaton b Φ( t, ˆ x ) = 12 ln (cid:18) − e x e x − (cid:19) . (51)Generalized Supergravity Equations are satisfied for background (50), dilaton(51) and vector field b J ( t, ˆ x ) = (cid:18) − e ˆ x (cid:19) ∂ ˆ x (52)obtained from (25). X-form corresponding to (51) and (52) b X ( t, ˆ x ) = e ˆ x (cid:0) e ˆ x − (cid:1) e x − ! d ˆ x (53)is closed so we can eliminate b J by gauge transformation. Dilaton then reads b Φ ′ ( t, ˆ x ) = 12 ln (cid:18) (2 + e ˆ x ) − e ˆ x (cid:19) (54)and satisfies beta function equations together with (50).14 .2.4 Plurality (1 | → ( ii | − ) Plurality given by matrix C (1 | → ( ii | − ) = D · C (1 | → (6 − | ii gives flat andtorsionless background b F ( t, ˆ x ) = − e ˆ x e ˆ x +1 3 e x ˆ x +4ˆ x e ˆ x +3 e x − e ˆ x ˆ x +2 e x ˆ x e ˆ x +2 e ˆ x ˆ x − e x ˆ x − e ˆ x e x (55)and dilaton b Φ( t, ˆ x ) = 12 ln (cid:18) e x − e ˆ x − e x + 1 (cid:19) . (56)Generalized Supergravity Equations are satisfied for (55), (56) and Killingvector field b J ( t, ˆ x ) = − ∂ ˆ x . (57)X-form corresponding to (56) and (57) b X ( t, ˆ x ) = 2 e ˆ x ( e ˆ x + 1) (3 e ˆ x − d ˆ x is closed and by gauge transformation to dilaton b Φ ′ ( t, ˆ x ) = 12 ln 1 − e ˆ x e ˆ x . (58)we get solution of beta function equations. It follows from the examples in Sections 4.1 and 4.2 that formulas (12) and(20) for Poisson–Lie transformations of dilatons and Killing vectors [5, 7,8, 9] are not universal in the sense that b Φ and b J together with transformedbackgrounds in general do not satisfy beta function equations nor GeneralizedSupergravity Equations. They work properly for transformations of isometricsigma models based on semi-abelian Manin triples ( d , g , a ) but not in othercases. 15e propose modification (25) of formula (20) giving vector fields whichtogether with dilatons given by formula (12) satisfy Generalized SupergravityEquations for all presented examples (and many other).From the form of NS-NS sector of Generalized Supergravity Equationsof Motion it is clear that knowledge of one-form X is important for theirsatisfaction, not separately dilaton Φ and vector field J . Therefore, besidethe Poisson–Lie transformation of tensor F it is sufficient to know only thetransformation of the form X to keep the Generalized Supergravity Equationssatisfied. The corresponding formula (26) was checked as well.In many examples the form X is closed so we can choose J vanishingby gauge transformation (22), and Generalized Supergravity Equations ofMotion become usual beta function equations. The same transformation canbe used to make J Killing vector field. Resulting dilatons then differ fromthose obtained from (12).
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