Chern-Simons--Antoniadis-Savvidy forms and standard supergravity
aa r X i v : . [ h e p - t h ] M a r Chern–Simons–Antoniadis–Savvidy forms and standardsupergravity
F. Izaurieta, ∗ P. Salgado, † and S. Salgado ‡ Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile
Abstract
In the context of the so called the Chern–Simons–Antoniadis–Savvidy (ChSAS) forms, we use themethods for FDA decomposition in 1-forms to construct a four-dimensional ChSAS supergravityaction for the Maxwell superalgebra. On the another hand, we use the Extended Cartan HomotopyFormula to find a method that allows the separation of the ChSAS action into bulk and boundarycontributions and permits the splitting of the bulk Lagrangian into pieces that reflect the particularsubspace structure of the gauge algebra. ∗ Electronic address: fi[email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION In Refs. [1–4] Antoniadis, Konitopoulos and Savvidy introduced a procedure to constructgauge invariant, background-free gauge forms. The integrals of these forms over the cor-responding space-time coordinates provides new topological actions that we have calledChern–Simons–Antoniadis–Savvidy (ChSAS) actions, which generalize the usual Chern–Simons theory. Of special interest are those which can be constructed in even dimensions.Using calculation methods for Free Differential Algebra (FDA) that allow the decompo-sition of p -forms in 1-forms developed in Ref. [5] and applied in Refs. [6–8], we construct afour-dimensional ChSAS supergravity action for Maxwell superalgebra.Following Ref. [9] it is found, in the context of the so called the ChSAS theory, asubspace separation method for the Lagrangian. The method is based on the iterativeuse of the generalized Extended Cartan Homotopy Formula, and allows one to separatethe action in bulk and boundary contributions, and systematically split the Lagrangian inappropriate reflection of the subspace structure of the gauge algebra. In order to apply themethod, one must regard ChSAS forms as a particular case of more general objects knownas generalized transgression forms.This work is organized as follows: in Section 2, we briefly review the principal aspects oftransgression and Chern–Simons–Antoniadis–Savvidy forms. In section 3, we use the calcu-lation methods for Free Differential Algebra (FDA) that allow the decomposition of p -formsin 1-forms developed in Ref. [5] and applied in Refs. [6–8] to construct a four-dimensionalChern–Simons–Antoniadis–Savvidy supergravity action for Maxwell superalgebra. Section4 presents the generalized Extended Cartan Homotopy Formula and shows how a subspaceseparation method that allows for a deeper understanding of the ChSAS Lagrangian can bebuilt upon it. We finish in Section 5 with conclusions and some considerations on futurepossible developments. II. CHERN–SIMONS–ANTONIADIS–SAVVIDY FORMS IN (2 n + 2) -DIMENSIONS The idea of extending the Yang–Mills fields to higher rank tensor gauge fields was usedin Ref. [1] in order to construct gauge invariant and metric independent forms in higher2imensions. These forms are analogous to the Pontryagin–Chern forms in Yang–Mills gaugetheory. These results were generalized in Refs. [2–4], where the authors found closed invariantforms similar to the Pontryagin–Chern forms in non-abelian tensor gauge field theory. Theseforms are based on non-abelian tensor gauge fields and are polynomials on the correspondingcurvature forms.A Lie algebra valued 1-form connection A can be written making more or less explicitthe dependence on the Lie algebra generator basis T a or the basis of 1-forms d x µ , A = A µ ⊗ d x µ = A aµ T a ⊗ d x µ . The same is true for the 2-form B gauge potential B = B µν ⊗ d x µ d x ν = B aµν T a ⊗ d x µ d x ν .The corresponding 2-form and 3-form “curvatures” are given by F = F µν ⊗ d x µ d x ν and H = H µνλ ⊗ d x µ d x ν d x λ respectively, where F = d A + A , H = D B = d B + [ A, B ] . (1)The curvatures F and H satisfy the Bianchi identitiesD F = 0 , D H + [ B, F ] = 0 . (2)The infinitesimal, non-abelian gauge transformations for the generalized gauge fields aregiven by δA = D ξ , δB = D ξ + [ B, ξ ] , (3)where ξ and ξ are a 0-form and a 1-form gauge parameters respectively [1]. Under thesegauge transformations, the curvatures transform as [2], δF = D( δA ) = [ F, ξ ] , (4) δH = D( δB ) + [ δA, B ] . (5)It may be of interest to note that we have used the definition of the conmutator fordifferential forms given by [ X, Y ] = XY − ( − pq Y X, where X is a p -form and Y is a q -form.In Refs. [1, 2] there were found closed invariant forms similar to the Pontryagin–Chernforms in non-abelian tensor gauge field theory. In particular, it was found that there exists3 gauge invariant metric-independent invariant in (2 n + 3)-dimensional space-timeΓ n +3 = h F n H i . (6) Chern–Weil theorem in the (2 n + 2) -dimensional case : the theorem ingredients are:( i ) Two Lie-algebra valued, 1-forms connections A and A . Their curvatures are given by, F = d A + A and F = d A + A , respectively. ( ii ) Two Lie-algebra valued, generalized 2-forms gauge fields B and B . Their generalized curvatures are given by H = d B + [ A , B ]and H = d B + [ A , B ] respectively ( iii ) In terms of these fundamental ingredients, it ispossible to define the differences Θ = A − A and Φ = B − B , and the interpolatingconnections A t = A + t Θ and B t = B + t Φ with 0 ≤ t ≤ . Their corresponding curvaturesare given by F t = d A t + A t , H t = D t B t = d B t + [ A t , B t ] , (7)which satisfy the conditionsdd t F t = D t Θ , dd t H t = D t Φ + [Θ , B t ] . (8) Theorem [8]:
Let A and A be two gauge connection 1-forms, and let F and F betheir corresponding 2-forms curvature. Let B and B be two gauge connection 2-forms andlet H and H be their corresponding curvature 3-forms. Then, the difference Γ (1)2 n +3 − Γ (0)2 n +3 is an exact formΓ (1)2 n +3 − Γ (0)2 n +3 = h F n H i − h F n H i = d T (2 n +2) ( A , B ; A , B ) , (9)where T (2 n +2) ( A , B ; A , B ) = Z d t (cid:0) n h F n − Θ H t i + h F nt Φ i (cid:1) (10)is what we call “Antoniadis–Savvidy transgression form”. A proof can be found in Ref. [8].Following the same procedure followed in the case of the Chern–Simons forms, we define the(2 n + 2)-Chern–Simons–Antoniadis–Savvidy form as C (2 n +2)ChSAS = T (2 n +2) ( A, B ; 0 ,
0) = Z d t h nAF n − t H t + BF nt i . (11)This result agrees with the expression found by Antoniadis and Savvidy in Refs. [1, 2]. Fromeq. ( ), we have for the n = 1 case [2], C (4)ChSAS = Z d t h AH t + F t B i = h F B i − d h AB i . (12)4t is interesting to notice that transgression forms (both, standard ones and the abovegeneralization) are defined globally on the spacetime basis manifold of the principal bun-dle, and are off-shell gauge invariant. Chern–Simons forms (both, standard ones and theAntoniadis–Savvidy generalization) are locally defined and are off-shell gauge invariant onlyup to boundary terms (i.e., quasi-invariants). III. CHERN–SIMONS–ANTONIADIS–SAVVIDY FORM FOR MAXWELL SU-PERALGEBRA
Now we will use this construction for the particular case of the Maxwell superalgebra, inorder to show the connection between eq. (12) and supergravity in D = 4. A. sM Maxwell superalgebra
The minimal Maxwell superalgebra sM in D = 4 is an algebra whose generators { P a , J ab , Z ab , ˜ Z ab , Q α , Σ α } satisfy the following commutation relation [10, 11][ J ab , J cd ] = η bc J ad + η ad J bc − η bd J ac − η ac J bd , [ J ab , P c ] = η bc P a − η ac P b , [ P a , P b ] = Z ab , [ J ab , Z cd ] = η bc Z ad + η ad Z bc − η bd Z ac − η ac Z bd , [ P a , Q α ] = −
12 ( γ a Σ) α , [ J ab , Q α ] = −
12 ( γ ab Q ) α , [ J ab , Σ α ] = −
12 ( γ ab Σ) α , h ˜ Z ab , Q α i = −
12 ( γ ab Σ) α , { Q α , Q β } = − h(cid:0) γ ab C (cid:1) αβ ˜ Z ab − γ a C ) αβ P a i , { Q α , Σ β } = − (cid:0) γ ab C (cid:1) αβ Z ab , h J ab , ˜ Z cd i = η bc ˜ Z ad + η ad ˜ Z bc − η bd ˜ Z ac − η ac ˜ Z bd , h ˜ Z ab , ˜ Z cd i = η bc Z ad + η ad Z bc − η bd Z ac − η ac Z bd , others = 0 . This algebra can be found by an S-expansion of osp (4 /
1) superalgebra. [10, 11].In order to write down a four dimensional ChSAS action, we start by expressing thegauge fields A and B at the base of Maxwell superalgebra.5o interpret the gauge field associated with a traslational generator P a as the vielbein, oneis forced to introduce a length scale ℓ in the theory. Since one can always chooses Lie algebragenerators T A to be dimensionless as well, the one-form connection fields A = A Aµ T A d x µ mustalso be dimensionless. However, the vielbein e a = e a µ d x µ must have dimensions of length ifit is related to the spacetime metric g µν through the usual equation g µν = e a µ e b ν η ab . Thismeans that the “true” gauge field must be of the form e a /ℓ , with ℓ a length parameter.Therefore, following Refs. [12], [13], the one-form gauge field A is given by A = 1 ℓ e a P a + 12 ω ab J ab + 12 k ab Z ab + 12 ˜ k ab ˜ Z ab + 1 √ ℓ ψ α Q α + 1 √ ℓ ξ α Σ α , where e a is identified as the 1 - form vierbein, ω ab is the 1 - form spin connection, k ab and ˜ k ab are extra antisymmetric bosonic 1-form fields, and ψ α , ξ α are fermionic 1-form fields. Thecorresponding 2-form curvature is given by F = 1 ℓ ˆ T a P a + 12 R ab J ab + 12 f ab Z ab + 12 ˜ f ab ˜ Z ab + 1 √ ℓ Ψ α Q α + 1 √ ℓ Ξ β Σ β , with ˆ T a = T a + 12 ¯ ψγ a ψ,R ab = d ω ab + ω ac ω cb ,f ab = D k ab + 1 ℓ e a e b + ˜ k ac ˜ k cb − ℓ ¯ ξγ ab ψ, ˜ f ab = D˜ k ab − ℓ ¯ ψγ ab ψ, Ψ α = D ψ α , Ξ β = D ξ β −
14 ˜ k ab ψ α ( γ ab ) βα − ℓ e a ψ α ( γ a ) βα . For the 2-form B , we can write B = B a P a + 12 B ab J ab + 12 β ab Z ab + 12 ˜ β ab ˜ Z ab + λ α Q α + χ α Σ α , where B a , B ab , β ab , ˜ β ab , λ α , χ α are 2-forms that we must determine. The corresponding3-form curvature is given by H = H a P a + 12 H ab J ab + 12 Θ ab Z ab + 12 ˜Θ ab ˜ Z ab + ˜ H α Q α + H α Σ α , H a = D B a − ℓ B ac e c + 1 √ ℓ ¯ ψγ a λ, (13) H ab = D B ab , (14)Θ ab = D β ab − B [ a | c k c | b ] − ℓ B [ a | e | b ] − ˜ β [ a | c ˜ k c | b ] − √ ℓ ¯ ψγ ab χ + 1 √ ℓ ¯ λγ ab ξ, ˜Θ ab = D ˜ β ab − B [ a | c ˜ k c | b ] − √ ℓ ¯ ψγ ab λ, (15)˜ H α = D λ α + 14 √ ℓ B ab ψ β ( γ ab ) αβ , (16) H α = D χ α + 14 √ ℓ B ab ξ β ( γ ab ) αβ − ℓ e a λ β ( γ a ) αβ + 12 √ ℓ B a ψ β ( γ a ) αβ −
14 ˜ k ab λ β ( γ ab ) αβ + 14 √ ℓ ˜ β ab ψ β ( γ ab ) αβ . (17)The problem now is to express the form B defined by the equations (13-17) in terms ofthe one-forms { e a , ω ab , k ab , ˜ k ab , ψ α , ξ α } of the Maxwell superalgebra. To express the 2-forms n B a , B ab , β ab , ˜ β ab , λ, χ o as the wedge product of the 1-forms { e a , ω ab , k ab , ˜ k ab , ψ α , ξ α } wefollow a procedure developed in Refs. [5, 6]. Imposing the ansatz B a = a ℓ ω ab e b + a ℓ k ab e b + a ℓ ˜ k ab e b + a ℓ ¯ ψγ a ψ + a ℓ ¯ ψγ a ξ + a ℓ ¯ ξγ a ξ, (18) B ab = b ℓ e a e b + b ω [ a | c k cb ] + b k ac k cb + b ω [ a | c ˜ k c | b ] + b k ac ˜ k cb + b ω ac ω cb + b ℓ ¯ ψγ ab ψ + b ℓ ¯ ψγ ab ξ + b ℓ ¯ ξγ ab ξ, (19) β ab = c ℓ e a e b + c ω [ a | c k c | b ] + c k ac k cb + c ω [ a | c ˜ k c | b ] + c k ac ˜ k cb + c ω ac ω cb + c ℓ ¯ ψγ ab ψ + c ℓ ¯ ψγ ab ξ + c ℓ ¯ ξγ ab ξ, (20)˜ β ab = d ℓ e a e b + d ω [ a | c k c | b ] + d k ac k cb + d ω [ a | c ˜ k c | b ] + d k ac ˜ k cb + d ω ac ω cb + d ℓ ¯ ψγ ab ψ + d ℓ ¯ ψγ ab ξ + d ℓ ¯ ξγ ab ξ, (21) λ α = f ℓ e a γ a ψ α + f ℓ e a γ a ξ α + f ω ab γ ab ψ α + f ω ab γ ab ξ α + f k ab γ ab ψ α + f k ab γ ab ξ α + f k ab γ ab ψ α + f k ab γ ab ξ α , (22) χ α = g ℓ e a γ a ψ α + g ℓ e a γ a ξ α + g ω ab γ ab ψ α + g ω ab γ ab ξ α + g k ab γ ab ψ α + g k ab γ ab ξ α + g k ab γ ab ψ α + g k ab γ ab ξ α , (23)where a , . . . , a , b , . . . , b , c , . . . , c , d , . . . , d , f , . . . , f , g , . . . , g are arbitrary constants,and introducing eqs. (18-23) in eqs. (13-17), when H a = H ab = Θ ab = ˜Θ ab = ˜ H α = H α = 0,7e find B a = a ℓ ¯ ψγ a ψ + a ℓ ¯ ψγ a ξ, (24) B ab = 0 , (25) β ab = c ℓ e a e b + c k ac ˜ k cb + c ℓ ¯ ψγ ab ξ + c ℓ ¯ ξγ ab ξ, (26)˜ β ab = d ℓ ¯ ψγ ab ψ + d ℓ ¯ ψγ ab ξ, (27) λ α = f ℓ e a γ a ψ α + f k ab γ ab ψ α , (28) χ α = g ℓ e a γ a ψ α + g ℓ e a γ a ξ α + g k ab γ ab ψ α + g k ab γ ab ξ α . (29)The fields given by eqs. ( - ) represent the most general solution that can be built fromthe fields { e a , ω ab , k ab , ˜ k ab , ψ α , ξ α } . Any choice of the constants represent a solution to theFDA. B. Chern-Simons-Antoniadis-Savvidy form
Using the invariant tensor found in Ref. [10] h J ab J cd i = α ǫ abcd , D J ab ˜ Z cd E = α ǫ abcd , D ˜ Z ab ˜ Z cd E = α ǫ abcd , h J ab Z cd i = α ǫ abcd , h Q α Q β i = 2 α ( γ ) αβ , h Q α Σ β i = 2 α ( γ ) αβ , being α , α and α dimensionless arbitrary independent constants, the Chern–Simons–Antoniadis–Savvidy Lagrangian L (4)ChSAS ≡ C (4)ChSAS is explicitly given by L (4)ChSAS = 14 ǫ abcd (cid:0) α R ab B cd + α (cid:0) R ab β cd + f ab B cd (cid:1)(cid:1) + 14 ǫ abcd α (cid:18) R ab ˜ β cd + 14 ˜ f ab B cd (cid:19) + α ˜ f ab ˜ β cd + 2 α √ ℓ Ψ α ( γ ) βα λ β + 2 α √ ℓ Ψ α ( γ ) βα χ β + 2 α √ ℓ λ α ( γ ) βα Ξ β . (30)8sing the FDA expansion given by eqs. ( - ), the Chern–Simons–Antoniadis–SavvidyLagrangian for the Maxwell algebra takes the form L (4)ChSAS = 14 ǫ abcd (cid:16) α R ab (cid:16) c ℓ e c e d + c k c f ˜ k fd + c ℓ ¯ ψγ cd ξ + c ℓ ¯ ξγ cd ξ (cid:17)(cid:17) + 14 ǫ abcd α R ab (cid:18) d ℓ ¯ ψγ cd ψ + d ℓ ¯ ψγ cd ξ (cid:19) + α ˜ f ab (cid:18) d ℓ ¯ ψγ cd ψ + d ℓ ¯ ψγ cd ξ (cid:19) + 2 ℓ / ( α f + α g ) Ψ α e a ( γ ) βα ( γ a ) γβ ψ γ + (cid:18) f α √ ℓ + g α √ ℓ (cid:19) Ψ α ˜ k ab ( γ ) βα (cid:0) γ ab (cid:1) γβ ψ γ + g ℓ α √ ℓ Ψ α ( γ ) βα e a ( γ a ) γβ ξ γ + g α √ ℓ Ψ α ( γ ) βα ˜ k ab (cid:0) γ ab (cid:1) γβ ξ γ − f ℓ α √ ℓ e a ψ β ( γ a ) αβ ( γ ) γα Ξ γ − f α √ ℓ ˜ k ab ψ β (cid:0) γ ab (cid:1) αβ ( γ ) γα Ξ γ . (31)From eq. (31) we can see that if c = d = f = f = g = g = 0, which are conditionsconsistent with the equations (18-23) and (13-17), we have that L (4)ChSAS is given by L (4)ChSAS = α c ǫ abcd R ab e c e d + 2 α g √ ℓ Ψ γ e a γ a ψ + α c l ǫ abcd R ab ˜ k c f ˜ k fd + (cid:18) α c α d (cid:19) lǫ abcd R ab ¯ ψγ cd ξ + α d lǫ abcd ˜ f ab ¯ ψγ cd ξ + α g l / Ψ γ ˜ k ab γ ab ψ. (32)Here it is necessary to notice that: (a) The first two terms contains the Einstein–Hilbert and the Rarita-Schwinger termsgiven by ǫ abcde R ab e c e d e e and Ψ γ e a γ a ψ respectively. (b) The following terms could be interpreted as non-linear couplings between the bosonicand fermionic ”matter” fields ˜ k ab , ξ , the Rarita-Schwinger field ψ and the curvature, wherethe parameter l can be considered as a kind of coupling constant.From eq. (32), we can see that when l ≪ , the Chern–Simons–Antoniadis–SavvidyLagrangian for the Maxwell superalgebra is given by L (4)ChSAS = α c ǫ abcd R ab e c e d + 2 α g √ ℓ Ψ γ e a γ a ψ, (33)where we can see that the Chern–Simons–Savvidy Lagrangian reproduces, except for nu-merical coefficients, the Lagrangian for standard supergravity.It is perhaps interesting to note that the conmutation relation [ P a , P b ] = Z ab depends onthe Z ab generators. The consequences on the Lagrangian of this non-zero bracket are related9o the gauge field k ab associated to Z ab . If we not consider the k ab dependence, or if we takea limit on the theory in which the gauge field k ab effects are not included, then it is notsurprising that the curvature term in the above Lagrangian looks like the standard gravityLagrangian. IV. THE EXTENDED CARTAN HOMOTOPY FORMULA IN (2 n + 2) -DIMENSIONS The Extended Cartan Homotopy Formula ( ECHF ) reads [14] Z ∂T r +1 l pt p ! π = Z T r +1 l p +1 t ( p + 1)! d π + ( − p + q d Z T r +1 l p +1 t ( p + 1)! π, (34)where, in this case, π represents a polynomial in the forms { A t , B t , F t , H t , d t A t , d t F t } whichis also an m -form on M and a q -form on T r +1 , with m ≥ p and p + q = r . The exteriorderivatives on M and T r +1 are denoted respectively by d and d t . The operator l t , calledhomotopy derivation, maps differential forms on M and T r +1 according to l t : Ω a ( M ) × Ω b ( T r +1 ) → Ω a − ( M ) × Ω b +1 ( T r +1 ) , and it satisfies Leibniz’s rule as well as d as d t . In our case, we will consider the polinomial π = h F nt H t i . This choice has the three following properties: (i) π is M -closed, i.e., d π = 0,(ii) π is a 0-form on T r +1 , and (iii) π is a (2 n + 3)-form on M . The allowed values for p are p = 0 , . . . , n + 3. The ECHF reduces in this case to Z ∂T p +1 l pt p ! π = ( − p + q d Z T p +1 l p +1 t ( p + 1)! π. (35)Since the three operators d, d t and l t define a graded algebra given by [14]d = 0 , d t = 0 , { d , d t } = 0 , (36)[ l t , d] = d t , [ l t , d t ] = 0 , (37)we have that the action of l t on { A t , B t , F t , H t , d t A t , d t F t } reads [14] l t A t = 0 , l t F t = l t (d A t + A t A t ) = (d l t + d t ) A t = d t A t , while the action of l t on B t and H t must be determined.10articular cases of (35) with π given by (6) which we review below, reproduce both theChern–Weil Theorem and the Triangle Formula. In fact, when p = 0 , we find that Eq.(35)takes the form, Z ∂T π = d Z T l t π, (38)where π = h F nt H t i and A t = A + t Θ , B t = B + t Φ. The left side of (38) is given by Z ∂T h F nt , H t i = Z d t h F nt , H t i = h F n , H i − h F n , H i , while for the right side we haved Z T l t h F nt , H t i = d (cid:26) n Z d t (cid:10) F n − t , Θ , H t (cid:11) + Z T h F nt , l t H t i (cid:27) , so that h F n , H i − h F n , H i = d (cid:26) n Z dt (cid:10) F n − t , Θ , H t (cid:11) + Z T h F nt , l t H t i (cid:27) . From the Chern–Weil theorem and B t = B + t Φ, we see that h F nt , l t H t i = h F nt , d t B t i , so that l t H t = d t B t . On the other hand, we have h F nt , l t (d B t + A t B t − B t A t ) i = h F nt , d t B t i , h F nt , ((d + A t ) ( l t B t ) + d t B t − ( l t B t ) A t ) i = h F nt , d t B t i , and therefore h F nt , D t l t B t i = d h F nt , l t B t i = 0 and l t B t = 0 . Summarizing, we can write l t B t = 0 and l t H t = d t B t . A. The subspace separation method
In this subsection we will show that the subspace separation method developed in Ref. [9]can be generalized to the case of Chern–Simons–Antoniadis–Savvidy formalism. This meansthat the so called triangle equation (46) splits the transgression form T (2 n +2) ( A , B ; A , B )into the sum of two transgressions forms depending on an intermediate connection A , B plus a exact form T (2 n +1) ( A , B ; A , B ; A , B ) shown in Eq. (43).When p = 1 we have that Eq. (35) is given by Z ∂T l t h F nt , H t i = − d Z T l t h F nt , H t i , (39)11here A t = t ( A − A ) + t ( A − A ) + A ,B t = t ( B − B ) + t ( B − B ) + B . The left side of (39) corresponds to an integral along the boundary of the simplex T =( A , B ; A , B ; A , B ) : Z ∂T l t h F nt , H t i = T (2 n +2) ( A , B ; A , B ) − T (2 n +2) ( A , B ; A , B )+ T (2 n +2) ( A , B ; A , B ) . (40)The right side of (39) is given byd Z T l t h F nt , H t i = d T (2 n +1) ( A , B ; A , B ; A , B ) , (41)where T (2 n +1) ( A , B ; A , B ; A , B ) = Z d t Z t d s (cid:8) n ( n − (cid:10) F n − t , ( A − A ) , ( A − A ) , H t (cid:11) + n (cid:10) F n − t , A , ( B − B ) (cid:11) + n (cid:10) F n − t , A , ( B − B ) (cid:11) (42)+ n (cid:10) F n − t , A , ( B − B ) (cid:11)(cid:9) . (43)In (43) we have introduced dummy parameters t = 1 − t and s = t , in terms of which A t reads A t = A + t ( A − A ) + s ( A − A ) . (44)Thus we have that the triangle equation is given by T (2 n +2) ( A , B ; A , B ) − T (2 n +2) ( A , B ; A , B ) + T (2 n +2) ( A , B ; A , B )= − d T (2 n +1) ( A , B ; A , B ; A , B ) , (45)or alternatively T (2 n +2) ( A , B ; A , B ) = T (2 n +2) ( A , B ; A , B ) + T (2 n +2) ( A , B ; A , B )+ d T (2 n +1) ( A , B ; A , B ; A , B ) . (46)We would like to stress that use of the Extended Cartan Homotopy Formula has allowed usto pinpoint the exact form of the boundary contribution T (2 n +1) ( A , B ; A , B ; A , B ) . A = 0 and B = 0 we obtain an expression that relates theform Antoniadi–Savvidy transgression form to two Chern-Simons-Savvidy forms and a totalderivative T (2 n +2) ( A , B ; A , B ) = C (2 n +2)ChSAS ( A , B ) − C (2 n +2)ChSAS ( A , B ) − d T (2 n +1) ( A , B ; A , B ; 0 , . (47) V. CONCLUDING REMARKS
In this Letter some features of the (2 n + 2)-dimensional transgressions and Chern–Simons–Antoniadis–Savvidy forms used as Lagrangians for supergravity theories were brieflyreviewed. The 2–form field B can be discomposed in terms of components of the 1-form A .It is performed in a self–consistent way by considering the generalization of Maurer–Cartanapproach to forms of higher order, i.e., free differential algebras, and by following the proce-dure used in Refs. [5–8]. The final result is a four-dimensional supergravity action, which isgauge quasi–invariant under the Maxwell superalgebra. These 4-dimensional results shownthat an interesting problem is to extract physical information from the (2 n + 2)-dimensionalLagrangian (11). A crucial step in this direction is the separation of the Lagrangian in a waythat reflects the inner subspace structure of the gauge algebra. This is specially interestingin the case of higher-dimensional supergravity, where superalgebras come naturally split intodistinct subspaces. Examples of the use of the method within the transgression/Chern–Simons–Antoniadis–Savvidy framework will be studied elsewhere. Acknowledgement 1
This work was supported in part by FONDECYT grants 1130653and 1150719 from the Government of Chile. One of the authors (SS) was suported by grants21140490 from CONICYT (National Commission for Scientific and Technological Re-search,spanish initials) and from Universidad de Concepci´on, Chile. [1] G. Savvidy, Phys. Lett. B (2010) 65.[2] I. Antoniadis, G. Savvidy, Eur. Phys. J. C (2012) 2140.[3] G. Savvidy, In. J. Mod. Phys. A (2014) 1450027.
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