(Super)conformal gravity with totally antisymmetric torsion
aa r X i v : . [ h e p - t h ] J a n (Super)conformal gravity with totallyantisymmetric torsion Riccardo D’Auria , Lucrezia Ravera , DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy.
Abstract
We present a gauge theory of the conformal group in four spacetime dimensions witha non-vanishing torsion. In particular, we allow for a completely antisymmetric torsion,equivalent by Hodge duality to an axial vector whose presence does not spoil the conformalinvariance of the theory, in contrast with claims of antecedent literature. The requirementof conformal invariance implies a differential condition on the aforementioned axial vectorwhich leads to a Maxwell-like equation in a four-dimensional curved background. Wealso give some preliminary results in the context of N = 1 four-dimensional conformalsupergravity in the geometric approach, showing that if we only allow for the constraintof vanishing supertorsion all the other constraints imposed in the spacetime approachare a consequence of the closure of the Bianchi identities in superspace. This paves theway towards a future complete investigation in the direction of a conformal theory ofsupergravity with a non-vanishing (super)torsion. E-mail: [email protected] ; [email protected] Introduction
In [1] it was shown that the locally scale invariant Weyl theory of gravity is the gauge theoryof the conformal group, where conformal transformations (conformal boosts) are gauged bya non-propagating gauge field. In that theory the authors adopted the formalism of [2, 3]to construct a quadratic Lagrangian with the curvatures associated with the conformalgroup in four spacetime dimensions. They claimed that in order to produce a conformallyinvariant theory in this setup it is necessary to set the torsion to zero. In this work, in contrast with this claim, we show that it is actually possible to con-struct a gauge theory of the conformal group in four spacetime dimensions with a non-vanishing torsion component where proper conformal transformations are gauged by anon-propagating gauge field (the Schouten 1-form field). In particular, we allow for a to-tally antisymmetric torsion, equivalent by Hodge duality to an axial vector, and still get aconformal gauge theory whose Lagrangian is quadratic in the curvatures of the conformalalgebra (as the one of [1], that is the same construction of [2, 3]). We explicitly show howto reproduce the Weyl Lagrangian in this framework in the presence of a non-vanishingcompletely antisymmetric torsion and study the field equations of the theory. Let us alsomention that for quadratic theories, in general, working in the first order or in the secondorder formalism for the spin connection one obtains different results. We will adopt thesecond order formalism, which will allow us to end up with a fourth order propagationequation for the graviton, the Lorentz connection being now torsionful.In this setup, invariance under conformal boosts (also known as proper, or special,conformal transformations) implies a differential condition on the axial vector torsion which,upon further differentiation, leads to a Maxwell-like equation in a four-dimensional curvedbackground. In the limit in which the torsion is set to zero we recover the conformal theoryof [1].The first part of the work will be devoted to study the purely bosonic gravitationaltheory. Subsequently, in view of a complete future investigation of the supersymmetricextension of this theory [4], we give some preliminary results we have obtained regardingconformal supergravity. We will adopt the geometric approach to supergravity (also called supergroup manifold approach or rheonomic approach ). As pioneering works on the structure of conformal supergravity at the linearized levelwe refer the reader to [7, 8]. The full conformal N = 1 supergravity theory in D = 3 + 1 Actually, in [1] the explicit form of the spin connection contains the dilaton, which gives a torsion tracecontribution. However, the latter can be consistently set to zero in the theory, as already observed in [1]and as we will also discuss in the present work. Therefore, let us refer to the theory of [1] as a torsion-freeone. For details on this formalism, see the original formulation in [5] and the pedagogical review [6]. N -extended conformal supergravity and its spectrum in four dimensionshave been recently obtained in Ref. [13]. In all these papers, together with the vanishing of(super)torsion, a set of additional constraints were also imposed. In particular in Ref. [12]the constraints were implemented by the use of Lagrange multipliers. The constraint ofvanishing supertorsion was justified arguing that only in this case the Lagrangian wouldhave been invariant under special conformal transformations. Since we will prove in thesequel of this work that, at least at the purely bosonic level, one can still recover invarianceunder (special) conformal transformations allowing for a non-vanishing axial vector torsion,we argue that something similar should presumably happen in the superconformal case.In view of future investigations in this direction, here we start a preliminary analysisat the level of Bianchi identities in the geometric approach, showing that, besides thevanishing supertorsion, all the aforementioned constraints can be directly obtained fromthe study of the Bianchi identities, just imposing the vanishing of the supertorsion (thatis the supersymmetric extension of the constraint imposed in [1], where torsion was indeedassumed to vanish).The paper is organized as follows: In Section 2 we give a brief review of the confor-mal setup in four spacetime dimensions. In Section 3 we develop the gauge theory of theconformal group with a non-vanishing completely antisymmetric torsion. Subsequently, inSection 4 we give some preliminary results regarding the extension to N = 1, D = 4 con-formal supergravity with vanishing supertorsion in the geometric approach. We concludeour work with some remarks and a discussion on future developments. In Appendix Asome useful formulas on gamma matrices in four dimensions are collected. The conformal group [15] O(4 ,
2) is locally isomorphic to SU(2 , T A = { J ab , P a , K a , D } , where we have de-composed the adjoint index A of the conformal algebra with respect to the Lorentz indices a, b, . . . = 0 , , , J ab are the Lorentz rotations, P a the spacetime translations, K a theconformal boosts, and D the dilatation (scale transformation). In our conventions the See also Ref. [14], where the authors used the geometric approach as in the present case, albeit in adifferent context, and the same conclusions can be reached after appropriate truncations. η ab has signature (+ , − , − , − ). Let us introduce the gauge 1-form fields ω ab (spin connection ), V a (vierbein), K a (special conformal 1-form field), D (dilaton gauge field), respectively dual to the vectorfields generators of the conformal algebra, namely ω ab ( J cd ) = 2 δ abcd , V a ( P b ) = δ ab , K a ( K b ) = δ ab , D ( D ) = 1 . (2.1)We can then write the corresponding curvatures, R ab ≡ R ab − V [ a ∧ K b ] ,T a ≡ D V a + D ∧ V a , T a ≡ D K a − D ∧ K a ,G ≡ d D + 2 V a ∧ K a , (2.2)where D = d − ω is the Lorentz covariant derivative and R ab = dω ab − ω ac ∧ ω cb (2.3)is the Riemann curvature. Setting the curvatures (2.2) to zero, the vanishing right-handsides define the Maurer-Cartan equations, describing the “vacuum” (ground state), dualto the commutator algebra of the vector fields generators { J ab , P a , K a , D } (as it is wellknown, the d -closure of the Maurer-Cartan equations coincides with the Jacobi identitiesof the algebra). For the sake of convenience, let us also defineˆ D V a ≡ D V a + D ∧ V a = dV a − ω ab V b + D ∧ V a , ˆ D K a ≡ D K a − D ∧ K a = dK a − ω ab K b − D ∧ K a , (2.4)where ˆ D denotes the Lorentz and scale covariant differential. The length-scale weight ofthe 1-forms and of their corresponding curvatures are[ ω ab ] = [ D ] = 0 , [ V a ] = 1 , [ K a ] = − . (2.5) Regarding our conventions, throughout the paper we will use rigid Latin indices a, b, . . . = 0 , , , µ, ν, . . . = 0 , , , p -forms in terms of the vierbeinbasis rather that in terms of differentials. For example, a generic 2-form will be expanded as F A = F Abc V b V c = F Aµν dx µ ∧ dx n , where V a = V aµ dx µ . This choice is convenient for the extension of thetheory to superspace using the geometric formalism where the p -forms are expanded in terms of the fullsupervierbein basis ( V a , ψ α ), ψ α being the gravitino 1-form. We will come back to a preliminary study ofconformal supergravity in Section 4. We call ω ab the spin connection antisymmetric in a, b , ω ab = − ω ba , which may (and in fact will) involvetorsion. We will generally omit writing of the wedge product between differential forms in order to lighten thenotation. R ab ≡ R ab − V [ a K b ] ,T a ≡ ˆ D V a , T a ≡ ˆ D K a ,G ≡ d D + 2 V a K a . (2.6)and the Bianchi identities obeyed by the curvatures (2.6) are D R ab + 4( T [ a K b ] − V [ a T b ] ) = 0 , ˆ D T a + R ab V b − GV a = 0 , ˆ DT a + R ab K b + GK a = 0 ,dG − T a K a + 2 V a T a = 0 , (2.7)where ˆ D T a ≡ D T a + D ∧ T a , ˆ DT a ≡ DT a − D ∧ T a . (2.8)The conformal gauge transformations read δω ab = D ε ab + 4 ε [ a K b ] − V [ a ε b ] K ,δV a = ˆ D ε a + ε ab V b − ε D V a ,δK a = ˆ D ε aK + ε ab K b − ε D K a ,δ D = dε D − ε a K a + 2 V a ε aK , (2.9)where ε ab , ε a , ε aK , and ε D are the Lorentz, translations, conformal boosts, and dilatationparameters, respectively.Let us mention that the theory whose Lagrangian we are going to consider (see eq. (3.2)in the following) is invariant under diffeomorphisms by construction, since it is written interms of differential forms, but it is not invariant under spacetime translations. This iswhat commonly happens in gravitational theories. Thus, it is not a true “gauge” theoryof the conformal group. However we shall adopt the terminology of “gauge theory of theconformal group” since it is widely used in the literature, keeping in mind that, in fact, we5ust have diffeomorphisms invariance rather than invariance under spacetime translations. Finally, let us also recall that the curvatures (2.6) can be expanded along the vierbeins,which are dual to the spacetime translation generators. This amounts to the requirementof having conformal symmetry of a theory defined on spacetime and it is therefore a naturalphysical request to have a conformal gravity theory within our approach. Indeed, as Lorentzand scale symmetries are an exact invariance of the Lagrangian (which we will introducein the following), the coset SU(2 , , ⊗ O(1 , K a K a can be expressed in terms of contractions of the Riemann tensor(more precisely, the Schouten tensor, as we will discuss in the sequel), the cotangent spaceis spanned in terms of the vierbein only.Therefore the aforesaid expansion of the curvatures along the vierbein basis reads R ab = R abcd V c V d ,T a = T abc V b V c , T a = T abc V b V c ,G = G ab V a V b . (2.10)We will now proceed with the development of a gauge theory of the conformal group witha non-vanishing torsion. We consider the same action introduced in [1], which is the only parity conserving quadraticaction that can be constructed from the curvatures (2.6) without dimensional constants, A = Z M L , (3.1) We recall that if we let the index A denote the coadjoint representation, an infinitesimal diffeomorphismof anholonomic parameter ε A = ε ρ µ Aρ on any gauge field of the algebra µ A can be written as δ ε µ A = D ε A + ı ε R A , where D is the covariant derivative in the coadjoint representation. Therefore, the diffeomorphisms ofthe gauge fields differ from the gauge translations by a term proportional to the contraction of the curvaturealong an infinitesimal translation ε a P a , where ε a is the infinitesimal parameter. In the supersymmetriccase the contraction of the supercurvatures along a supersymmetric generator is also in general differentfrom zero. Therefore, in the superconformal case the superspace translations correspond to supersymmetrytransformations. L = R ab ∧ R cd ǫ abcd (3.2)is the Lagrangian 4-form and M is the four-dimensional spacetime.Let us first recall the well known fact that the variation of the action with respectto the special conformal 1-form K b gives an algebraic equation for the special conformalgauge field K a . Indeed, varying the Lagrangian (3.2) with respect to K b we obtain thefield equations − V a R cd ǫ abcd = 0 , (3.3)which imply, using the expansion along the vierbein basis (2.10), R = 0 , ˇ R ab = 0 , (3.4)where R = R abab and ˇ R ab = R cacb . Taking (3.4) together with the definition of R ab in (2.6)and writing K a = K ab V b , one gets K ab = S ab , (3.5)being S ab the Schouten 0-form tensor defined in four-dimensional spacetime as S ab ≡ (cid:18) ˇ R ab − η ab R (cid:19) , (3.6)where R = R abab and ˇ R ab are the scalar curvature and the Ricci tensor of ω ab , respectively.Notice that, in the presence of a non-vanishing torsion, ω ab also includes a contorsioncomponent implying that the Schouten tensor S ab has a non-vanishing antisymmetric part, S [ ab ] = ˇ R [ ab ] , given entirely in terms of torsion.Thus, we have obtained an algebraic equation for the non-propagating gauge field K ab ,namely eq. (3.5), which tells us that K ab corresponds to the Schouten tensor. This is a wellknown fact (see for instance [1]). However, some comment is in order on this point, andwe make it in the following. In particular, we will show that the fact that K ab correspondsto the Schouten can be actually deduced directly from a vacuum analysis and we will alsogive the irreducible decomposition of R abcd which will be useful in the sequel. Let us briefly discuss, before proceeding with our main results, that already at the vacuumlevel, as in fact expected, one can show that the field K a cannot be anything other thanthe Schouten 1-form, S a = S ab V b . This can be shown by taking into account the irreducibledecomposition of the Riemann tensor R abcd (here allowing also for the presence of torsion,see, for instance, [16] for details). Regarding the number of components, in four spacetime7imensions we have dim (cid:0) R abcd (cid:1) = 6 × ⊕ ⊕
1, corresponding in terms of the SL(4)representations to the dimensions of the following Young diagrams: ⊗ = ⊕ ⊕ (3.7)Decomposing the SL(4) representations with respect to SO(1 ,
3) in terms of their trace-less plus trace parts we find six irreducible pieces (irrepses): −→ ˚ ⊕ ˚ ⊕ • (3.8) −→ ˚ ⊕ ˚ (3.9) −→ (3.10)where the small ring on top of the diagrams on the right-hand sides means that the corre-sponding representation is traceless, while the bullet denotes the scalar representation.The three irrepses on the right-hand side of eq. (3.8) correspond to the 10-dimensionalWeyl tensor W abcd , the 9-dimensional traceless symmetric Ricci tensor ˚ˇ R ( ab ) , and the scalarcurvature R , respectively, and, using the nomenclature of Ref. [16], are called WEYL +RICSIMF + SCALAR. The two irrepses on the right-hand side of eq. (3.9) correspond tothe 9-dimensional tensor which has the same number of degrees of freedom of a symmetrictraceless tensor, plus the 6-dimensional traceless antisymmetric Ricci tensor ˇ R [ ab ] , shortlyreferred together to as PAIRCOM and RICANTI, respectively. Finally, on the right-handside of eq. (3.10) we have a pseudo-scalar Hodge-dual to R [ abcd ] denoted as PSSCALAR.Writing R abcd = 12 ( R abcd + R cdab ) | {z } WEYL + RICSIMF + SCALAR ⊕ (cid:2) ( R abcd − R cdab ) − R [ abcd ] (cid:3)| {z } PAIRCOM + RICANTI ⊕ R [ abcd ] | {z } PSSCALAR , (3.11)the three “underbraced” expressions correspond to the left-hand sides of the three eqs.(3.8), (3.9), and (3.10), respectively. We enumerate the various representatios writing R abcd = X i =1 R ab ( i ) | cd = R ab (1) | cd | {z } WEYL [10] + R ab (2) | cd | {z } PAIRCOM [9] + R ab (3) | cd | {z } PSSCALAR [1] + R ab (4) | cd | {z } RICSIMF [9] + R ab (5) | cd | {z } RICANTI [6] + R ab (6) | cd | {z } SCALAR [1] , (3.12) The corresponding representation is commonly referred to as associated . R ab (2) | cd , R ab (3) | cd , and R ab (5) | cd are non-vanishing only in the pres-ence of torsion (so that they are given in terms of torsion and its derivatives).Now, we can exploit (3.12) in the vacuum of our theory, given by the vanishing right-hand side of (2.6), In particular we have R ab − V [ a K b ] = 0 , (3.13)that is R abcd − δ [ a [ c K b ] d ] = 0 . (3.14)Using (3.12) and observing that the second term in (3.14) can be written only in terms ofthe irreducible pieces RICSYMF, SCALAR, and RICANTI, one can prove that K ab mustcoincide with the Schouten tensor S ab (that is, K a = S ab V b ), the components of the latterbeing defined in (3.6). Therefore, the Maurer-Cartan equations obtained from (2.6) takethe following form: R ab (2) ≡ R ab (2) | cd V c V d = 0 , R ab (3) ≡ R ab (3) | cd V c V d = 0 ,W ab ≡ R ab (1) = R ab − V [ a S b ] = 0 ,T a ≡ ˆ D V a = 0 , T a ≡ ˆ D S a = 0 ,G ≡ d D + 2 V a S a = 0 , (3.15)where R ab (2) and R ab (3) are the PAIRCOM and PSCALAR 2-forms, respectively, and W ab = W abcd V c V d , being W abcd the Weyl tensor.One could then go out of the vacuum switching on the curvatures associated with theMaurer-Cartan equations (3.15) and write a quadratic Lagrangian in terms of these field-strengths. We note that in the case of vanishing torsion ( T a = 0 ⇒ R ab (2) = R ab (3) = 0) suchLagrangian reads L W = W ab ∧ W cd ǫ abcd , (3.16)where W ab = W ab ( ω ) with ω = ω ( D , V ) (note that we still have R ab (5) | cd = 0, due to thepresence of D ). The Lagrangian (3.16) coincides with the Lagrangian given in Ref. [1].In that paper the authors set the torsion T a equal to zero right from the beginning, andplugging back into (3.2) the on-shell expression (3.5) for K ab , they recover the same Weyl Indeed, tracing the b, d indices of both terms on the left-hand side of eq. (3.14) one easily recover K ab = S ab . T a is zero, withthe theory of Ref. [1]. In fact, eq. (3.5) shall be interpreted directly as a consequence ofthe structure of the vacuum of the theory quadratic in the Weyl tensor, which is indeedthe conformal theory we are going to focus on.In the sequel we will show that, remarkably, the Lagrangian we will develop describinga gauge theory of the conformal group with a non-vanishing torsion is formally identical tothe Lagrangian (3.16) provided the curvatures are constructed from a torsionful connection.In view of this, let us proceed by first showing that it is still possible to get conformalinvariance of the theory in the presence of a non-vanishing T a . In other words, we aregoing to prove that the constraint of vanishing torsion introduced in [1] to get a conformallyinvariant theory can be actually relaxed. The aim of this section is to see whether a non-vanishing T a is allowed in a “gauge” theoryof the conformal group. In particular, we will allow for a totally antisymmetric torsion,equivalent by Hodge duality to an axial vector. We will show that the requirement ofconformal invariance of the Lagrangian (3.2) constructed with the curvatures (2.6) can bestill fulfilled provided we require the vanishing of R ab (2) , R ab (3) , and R ab (5) (notations for theirrepses of R ab are the same as for R ab ). This will imply a differential condition on thecompletely antisymmetric part of the torsion, namely for the aforesaid Hodge-dual axialvector. Upon use of the on-shell conditions (3.4) implying (3.5), the theory will formally reproduce the same Lagrangian (3.16), albeit with a torsionful spin connection. Let us show the above explicitly. The Lagrangian (3.2) is clearly scale invariant, as R ab has zero scale weight. Nevertheless, the invariance under conformal boosts can beachieved only in a non-trivial way. In particular, one can prove that some constraints onthe curvatures (2.6) have to be imposed in order for (3.2) to be invariant under properconformal transformations on spacetime. Indeed, to recover invariance of (3.2) underconformal boosts, performing the variation δK d = ˆ D ε dK (see (2.9)), we must have (cid:0) R ab ǫ abcd T c (cid:1) ε dK = 0 , (3.17)that is, using (2.10), (cid:0) R ablm T cpq ǫ lmpq ǫ abcd Ω (4) (cid:1) ε dK = 0 , (3.18) Note that the same Lagrangian can be also obtained by directly gauging the Maurer-Cartan equations(3.15), that is switching on the corresponding curvatures going out of the vacuum, in the presence of acompletely antisymmetric torsion, and with the aforementioned constraints. (4) is the four-dimensional volume element defined as Ω (4) ≡ − ǫ abcd V a V b V c V d .In [1] the authors claimed that (3.2) results to be invariant under proper conformal gaugetransformations only if T a = 0 (the vanishing of R ablm being not considered as it wouldtrivialize the theory). Actually, this is not the case, as we will show in the sequel.In order to explain our claim in detail we need the irreducible decomposition of thetorsion in four spacetime dimensions (see, for instance, [16–18]). In four dimensions thetorsion tensor T abc has 24 = 20 ⊕ ⊗ = ˚ ⊕ ⊕ (3.19)whose dimensions are 16, 4, and 4, respectively, corresponding to the decomposition T abc = T ˚ b ac + 23 δ a [ b t c ] + T [ abc ] . (3.20)In the following we will denote the 16-dimensional representation as a tensor Z abc , that is T ˚ b ac = Z abc , (3.21)and the antisymmetric representation T [ abc ] as the axial vector ˜ t d , namely T [ abc ] = − ǫ abcd ˜ t d , (3.22)while t a appearing in the torsion trace part is an ordinary vector. Inserting the abovedecomposition of the torsion into (3.18), the latter becomes (cid:20) R ablm (cid:18) δ cp t q − ǫ cpqr ˜ t r + Z cpq (cid:19) ǫ lmpq ǫ abcd (cid:21) ε dK = 0 . (3.23)The necessary condition given by (3.23) consists in a set of four algebraic equations (recallthat ε dK is arbitrary) in the curvatures R ab and T abc for which we are now going to examinein detail some particular solutions.We first observe that a sufficient constraint on the torsion to have conformal boostsinvariance is ˜ t a = Z abc = 0. Indeed, in this case (3.23) yields (cid:0) − Rt a + 2 ˇ R ba t b (cid:1) ε aK = 0 −→ Rt a − R ba t b = 0 , (3.24)which, for a non-vanishing torsion trace t a , has as a particular solution R = 0 , ˇ R ab = 0 . (3.25)The latter constraints coincide with the equations that one obtains when varying the La-grangian (3.2) with respect to K a (that is, when going on-shell for K a ), namely with (3.4).11owever, let us recall here that the torsion trace t a , even if perfectly allowed, as we havejust seen that it does not spoil the conformal invariance of the theory, can be actually setto zero in a consistent way (see e.g. [19] for details). Indeed, one can easily verify thatthe torsion is invariant under a shift of the dilaton D = D a V a by a parameter X a , namely D a → D ′ a = D a + X a , provided that the spin connection ω ab | m = ω ab | µ V µm transforms as ω ab | m → ω ′ ab | m = ω ab | m − δ m [ a X b ] , and with the choice X a = t a the torsion trace getsreabsorbed into the dilaton (we have D ′ a = D a + t a , and D ′ a will be again renamed as D a in the following). Thus, from now on we set t a = 0 . (3.26)It follows that the necessary condition for conformal invariance (3.23) becomes (cid:20) R ablm (cid:18) Z cpq − ǫ cpqr ˜ t r (cid:19) ǫ lmpq ǫ abcd (cid:21) ε dK = 0 . (3.27)A simple solution of eq. (3.27) can be found assuming Z cpq = 0 . (3.28)Using (3.28), eq. (3.27) becomes2 ǫ bdlm R adlm ˜ t a − ǫ cdlm R cdlm ˜ t b = 0 . (3.29)We will now show that (3.28) and the ensuing condition in (3.29) lead to intriguing physicalconsequences on the surviving field ˜ t a . Indeed, for ˜ t a = 0 a possible solution of (3.29) is R a [ bcd ] = 0 , R [ abcd ] = 0 . (3.30)We shall focus on this particular solution. The latter implies R ab (2) | cd + R ab (5) | cd = 0 , R ab (3) | cd = 0 . (3.31)Furthermore, recalling eq. (3.4), namelyˇ R ab = R = 0 ←→ R ab (4) | cd = R ab (5) | cd = R ab (6) | cd = 0 , (3.32) Here let us mention that, in fact, using (3.28) into (3.27) the latter boils down to R ablm ˜ t r ǫ cpqr ǫ lmpq ǫ abcd = 0 , which can be simplified by contracting either first ǫ cpqr ǫ lmpq and then the result with ǫ abcd , or first ǫ cpqr ǫ abcd and then the result with ǫ lmpq . The respectively obtained equations may appear different at first sight, butexploiting the symmetry properties of the irrepses of R abcd one can verify that they are actually equivalent,both exhibiting, in particular, (3.30) as a possible solution. R ab in (2.6), one can easily realize thatwe are left with R ab (2) | cd = 0 ⇒ R ab (2) | cd = 0 ,R ab (3) | cd = 0 ⇒ R ab (3) | cd = 0 ,R ab (4) | cd = 0 , R ab (5) | cd = 0 , R ab (6) | cd = 0 , (3.33)together with R ab (4) | cd + R ab (5) | cd + R ab (6) | cd ≡ δ [ a [ c S b ] d ] ,R ab (1) | cd = R ab (1) | cd ≡ W abcd . (3.34)Hence, since now we have R abcd = R ab (1) cd = W abcd ≡ R abcd − X i =2 R ab ( i ) | cd = R abcd − δ [ a [ c S b ] d ] , (3.35)we may write R ab = W ab ≡ R ab − V [ a S b ] = R ab − R [ a | c V c V b ] + 13 R V a V b , (3.36)which is formally identical to the torsionless W ab , but now ω ab contains a torsion part.In conclusion, we have recovered invariance under conformal boosts of (3.2) solving thenecessary condition for conformal invariance (3.27) under the assumption Z abc = 0, theonly non vanishing part of the torsion T a being given by T a = T abc V b V c = − ǫ abcd ˜ t d V b V c . (3.37)Inserting the actual form (3.37) into eq. (2.6) we are led to ω ab | m = ω ab | µ V µm = ˚ ω ab | m − η m [ a D b ] − ǫ abmc ˜ t c , (3.38)where ˚ ω ab | m = ˚ ω ab | µ V µm and the last term in (3.38) is the contribution due to the contorsionterm. Moreover, from the variation of the torsion definition in (2.6), we now have δω ab | m = (cid:16) δ lm δ q [ a δ pb ] + δ qm δ p [ a δ lb ] − δ pm δ l [ a δ qb ] (cid:17) ˆ D q ( δV p ) l − η m [ a δ cb ] δ D c − ǫ abmc δ ˜ t c − ǫ abpc ˜ t c δ lm ( δV p ) l , (3.39)and one can verify that the transformations of the fields in (3.39) are such that the variationof ω ab | m under dilatations and conformal boosts is the same whether one determines it from Recall also that ˚ ω ab | µ = (cid:0) f λ | µν + f ν | λµ − f µ | νλ (cid:1) V λa V νb , with f λ | µν = V kλ ∂ [ µ V cν ] η ck . Thus, the Lagrangian (3.2) remainsscale and proper conformal invariant if ω = ω ( V, D , ˜ t ) as given in (3.38).If we now substitute (3.36) into (3.2), we obtain a Lagrangian that is formally identicalto (3.16) but involving torsion, namely L = W ab W cd ǫ abcd = − W abcd W abcd Ω (4) = R ab R cd ǫ abcd − (cid:18) ˇ R ab ˇ R ba − R (cid:19) Ω (4) , (3.40)the torsion being now hidden in the torsionful spin connection.Furthermore, at the level of the theory (3.2) the dilaton can be consistently eliminated.Indeed, also in the present case (as already pointed out in [1] in the case of vanishing torsion)the kinetic term for D does not contribute, since in (3.40) we have the combination ˇ R ab ˇ R ba rather than ˇ R ab ˇ R ab (whose presence would instead yield such kinetic term for the dilaton),which makes a difference because ˇ R ab ( ω ) is not symmetric. A further check of the non-propagating nature of the dilaton can be ascertained from its equation of motion. Indeed,even if we retain D in the Lagrangian, one can verify that its equation of motion is actuallythe trivial identity. With these arguments, one may set D = 0 (3.41)from the start, but it is not immediately obvious how the Lagrangian remains invariant inthis case since now the variation of ω ab | m under dilatations and conformal boosts determinedfrom (3.39) is not the same as the one determined from the gauge prescription (2.9) anymore.This problem was solved in Ref. [1] (for the torsion-free theory) showing that the additionalterms present in the δω ab variation give a vanishing contributuion. The same mechanismholds true in our case. Indeed, when varying the Lagrangian we get an additional variation δ ′ L = δ L δω δω ′ , where δω ′ is the difference between the gauge variation of ω and the variationfound from the explicit form of ω = ω ( V, ˜ t ). In particular, we get δ ′ L = − (cid:18) ǫ clnr ˜ t r K dp − δ cl T dnp (cid:19) ǫ lnpq ǫ qbcd ξ b ⇒ δ ′ L = 16 (cid:18) ǫ blnp K ln ˜ t p + 4 T lbl (cid:19) ξ b , (3.42)where ξ b is either ∂ b ε D for dilations or 2 ε cK for conformal boosts. However, from theBianchi identity of R ab , which yields, in particular, T ala = − ǫ labn K ab ˜ t n , one can see Indeed, under dilatations we have δ ε D V pl = − ε D δ pl , δ ε D D c = ∂ c ε D = ˆ D c ε D , and δ ε D ˜ t c = ε D ˜ t c ,which plugged into (3.39) lead to δ ε D ω ab | m ( V, D , ˜ t ) = 0 (in particular, the torsion contributions canceleach other out), reproducing the same result that can be obtained from the gauge prescription (2.9).Analogously, under conformal boosts one has δ ε K V pl = 0, δ ε K D c = 2 ε K | c , and δ ε K ˜ t c = 0, implying δ ε K ω ab | m ( V, D , ˜ t ) = − η m [ a ε K | b ] , which is the same variation that one can obtain from (2.9). δ ′ L in (3.42) vanishes identically. Hence, one can set the dilaton to zero withoutspoiling the conformal invariance of the theory, and (3.40), that is Weyl gravity with anon-vanishing axial vector torsion, results to be the gauge theory of the conformal groupwith non-vanishing torsion.Having eliminated the dilaton, we are left with (3.36) and (3.40) where now the dilatoncontributions vanish. In particular, now we have G = 2 V a S a ⇒ G ab = ˇ R [ ab ] . (3.43)Recalling that ˇ R [ ab ] is a function of ˜ t a only and, in particular, through (3.38) we haveˇ R [ ab ] = − ǫ abcd D c ˜ t d , (3.44)eq. (3.43) becomes G ab = ˇ R [ ab ] = − ǫ abcd D c ˜ t d −→ ˇ R [ ab ] = − ⋆ T ab , (3.45)where the star symbol denotes the Hodge duality operator and we have defined the field-strength T ab ≡ D [ a ˜ t b ] . (3.46)We note that this result directly follows from the conformal invariance of the theory.Furthermore, let us observe that the conformal invariance constraints in (3.30) imply R a [ bcd ] = − η a [ d S bc ] −→ R a [ bcd ] = − η a [ d ˇ R bc ] , (3.47) ǫ abcd R abcd = 0 . (3.48)We now show that eqs. (3.47) and (3.48) imply a differential constraint on the axial vectorpart of the torsion. Indeed, from eq. (3.45), recalling that now we have ω = ω ( V, ˜ t ), thatis the spin connection also involves a contorsion part, one can show that (3.48) reduces to D a ˜ t a = 0 . (3.49)Then, in (3.47) we express both sides of the equation in terms of the torsion by exploitingthe fact that now the curvature and antisymmetric Ricci tensors are given entirely interms of the totally antisymmetric torsion. Indeed, recall that the connection ω ab | c involves,besides the usual Riemannian part, also a contorsion term K abc related to the completelyantisymmetric torsion as follows: V b K ab = T a ⇒ K abc = − ǫ abcd ˜ t d . (3.50)15s a consequence, using (3.49) into the explicit form of (3.47), one is left with D ( a ˜ t b ) = 0 . (3.51)Hence, from (3.30) we obtain eqs. (3.49) and (3.51) for the axial vector torsion ˜ t a . These im-ply, together, as can be proven by further differentiation and using the fact that [ D a , D b ] ˜ t m = − R mnab ˜ t b − T nab D n ˜ t m = − R mnab ˜ t n + ǫ abnd ˜ t d D n ˜ t m , (cid:3) ˜ t b − R ab ˜ t a − ǫ abcd ˜ t a T cd = 0 −→ (cid:3) ˜ t b − R ab ˜ t a + 4 ˇ R [ ab ] ˜ t a = 0 , (3.52)where (cid:3) denotes the covariant d’Alambertian with torsion, (cid:3) ˜ t b ≡ D a D a ˜ t b .Thus, as we can see from (3.52), in our theory the axial vector torsion obeys (3.52), andthis follows directly from the requirement of invariance of the Lagrangian under conformalboosts. Eq. (3.52) can be regarded as a Maxwell-like equation in a curved four-dimensionalbackground. Note, however, that this is not an equation of motion derived from theLagrangian, but just the result of having required the conformal invariance in the presenceof an axial vector torsion.As a last comment let us observe that, taking into account all the results obtained tillnow, the Bianchi identities (2.7) can be rewritten asBianchi for R ab ( R ab = W ab ) : D W ab + 4 (cid:0) T [ a S b ] − V [ a C b ] (cid:1) = 0 ←→ DR ab = 0 , Bianchi for T a : D T a + W ab V b − GV a = 0 , Bianchi for T a ( T a = C a ) : D C a + R ab S b = 0 , Bianchi for G : dG − T a S a + 2 V a C a = 0 , (3.53)where we encounter the vector valued Cotton 2-form defined as C a ≡ D S a −→ C a | bc V b V c = −D c S ab V b V c + S al T lbc V b V c = −D c S ab V b V c − S al ǫ lbcd ˜ t d V b V c . (3.54)In the case in which the torsion vanishes we recover the properties that the Schouten tensoris symmetric and that the completely antisymmetric and trace parts of the Cotton tensorare zero.As a final check of the consistency of the theory we can check from the previous equa-tions that the Bianchi identities are identically satisfied. Indeed analyzing the torsionBianchi in (3.53) by taking into account also (3.45) together with (3.49) and (3.51) (recallthat (3.45), (3.49), and (3.51) come from conformal invariance of the theory, not from theanalysis of the field equations), we get 0 = 0. One can prove, with some algebraic manip-ulation and making use of (3.52), coming from the requirement of conformal invariance,16hat the same happens for the Bianchi of G . The Bianchi for ˜ T a becomes the CottonBianchi (for a connection with torsion) since ˜ T a coincides with the Cotton 2-form, whilethe Bianchi identity for R ab is the Bianchi for the Weyl 2-form W ab and simply leads to R ab = 0, the left-hand side of the latter being identically zero for any connection (herewith a non-vanishing torsion). One can thus see that the Bianchi identities are, as expected, true identities, and theydo not add any additional constraint to the theory. This result (3.53) was expected alsofrom the vacuum analysis we have previously done.
Since we are adopting the second order formalism for ω , that is ω = ω ( V, D , ˜ t ), in particularby fixing the form of the torsion, we shall vary the Lagrangian (3.40) with respect to theindependent fields V and ˜ t . As we will see in a while, in this setup we get a propagation equation of the gravitonthat has four derivatives ( ∂ V ), as in the case with vanishing torsion, which is indeedexpected for a conformal gravity theory.Now, let us observe that the following applies to our analysis: δ Φ i L = δ L δω δωδ Φ i δ Φ i + δ L δ Φ i δ Φ i , i = 1 , , Φ = ˜ t , Φ = V , (3.55)schematically. The term δ L δ Φ i δ Φ i corresponds to the explicit variation of the various fieldsin the Lagrangian. Therefore, we may start by computing δ ω L = δ L δω δω , where L is thetorsionful Weyl Lagrangian (3.40). Recalling eq. (3.36) and using δ ω R ab = D ( δω ab ) = D c ( δω ab ) d V c V d ,δ ω (cid:0) R abcd δ db (cid:1) = D c ( δω ab ) b ,δ ω (cid:0) R abcd δ db δ ca (cid:1) = D a ( δω ab ) b , (3.56)after partial integration and using the fact that the Weyl tensor is completely traceless, wefind δ ω L = 8 δω ab | m (cid:0) D l W ablm (cid:1) Ω (4) , (3.57) The Weyl Bianchi written in the form DR ab = 0 and the Cotton Bianchi look formally the same as inthe case of vanishing torsion (see for instance [20]). In principle the spin connection ω ab has a dilaton dependence, but, as we have previously discussed,the dilaton field D can be eliminated from the theory. Actually, as one can easily verify, even if D isallowed to appear into the Lagrangian, exploiting the explicit form for the variation of ω ab (3.39) and thefact that the Weyl tensor is traceless, its dynamics is trivial, as expected. D b Ω (4) V b = 23! ǫ defg T dbk V b V k V e V f V g = − · ǫ dbkq ˜ t q ǫ defg ǫ kefg V b Ω (4) = 0 , (3.58)that is D b Ω (4) = 0 . (3.59)Regarding the variation with respect to ˜ t c , one finds that the explicit variation doesnot contribute. In fact one obtains δ L δω δωδ ˜ t c δ ˜ t c = − ǫ abmc δ ˜ t c (cid:0) D l W ablm (cid:1) Ω (4) = 0 , (3.60)which vanishes identically since antisymmetrization of W abcd on three indices gives identi-cally zero. Concerning the explicit variation with respect to V , here one can prove that we have δ L ( δV p ) l ( δV p ) l = − δV p ) l (cid:18) R ac δ cm η pb − R ap η bm − R η am η bp (cid:19) W ablm Ω (4) = − δV p ) l ˇ R ac W aplc Ω (4) , (3.61)where we have also observed that the second and third term inside the round brackets in thefirst line give a vanishing contribution since they imply a tracing of the Weyl tensor. Then,using (3.39), (3.57) (and performing a partial integration), and (3.61), together with thesymmetry properties of the Weyl tensor and eq. (3.59), after some algebraic manipulationwe find that the field equation δ V L = 0 reads( δV p ) l (cid:18) D q D t W tpql −
12 ˇ R ac W aplc − ǫ abpc ˜ t c D t W abtl (cid:19) = 0 , (3.62)namely D q D t W tpql −
12 ˇ R ac W aplc − ǫ abpc ˜ t c D t W abtl = 0 . (3.63)Notice that the p, l trace of the latter identically vanishes, due to the symmetry propertiesof the Weyl tensor. The first two terms in (3.63) are formally the same as in the absence oftorsion, thus giving a fourth order equation for the vierbein. At linearized level the kineticterm is actually the same as in the absence of torsion, while at higher level the presence ofcontorsion in the spin connection gives higher order corrections. The symmetry properties of the Weyl tensor read W abcd = − W abdc = − W bacd ,W abcd = W cdab , W a [ bcd ] = 0 , W [ abcd ] = 0 . Note that they hold true also in the presence of torsion, since the Weyl tensor is an irrep of SO(1 , t a is set to zero we recover the conformal theory of [1]. In this section we give some preliminary results concerning conformal N = 1, D = 4 su-pergravity in the geometric approach. As we have already mentioned, in the literature,besides vanishing supertorsion, some constraints have been implemented to recover su-perconformal invariance (the same constraints have been implemented in [12], within thegeometric approach, through Lagrange multipliers).Here we start a preliminary analysis at the level of Bianchi identities using the geometricapproach, showing that all the aforementioned constrained can be directly obtained fromthe study of the Bianchi identities, just imposing the vanishing of the supertorsion (whichcan be viewed as the direct supersymmetric extension of the constraint of vanishing torsionimposed in [1]). For the benefit of the reader and in order to establish our formalism let us just recallthe main basic points of the geometric approach to supergravity.The gauge fields are now super 1-forms in superspace that can be expanded along thesupervierbein ( V a , ψ α ), with α = 1 , . . . , ψ α being the gravitino 1-form (note thatin the geometric approach the superfields are never expanded in terms of the Grassmanncoordinates). Analogously, the supercurvatures are 2-forms which can be expanded alongbasis of 2-forms, namely R A = R Aab V a V b + R Aaα V a ψ α + R Aαβ ψ α ψ β , (4.1)where R Aaα and R Aαβ are the outer components of R A , while R Aab are the inner ones. The important point is that, both in the Lagrangian approach as well as in the Bianchiidentities approach, it turns out that all the outer components of the curvatures can beexpressed algebraically in terms of the inner ones, thus allowing for the elimination of the Let us observe that, as we have previously mentioned, the same eqs. (3.52) and (3.63) can be obtainedby gauging (3.15) and implementing the constraints we have presented in our analysis in order to recoverconformal invariance. For the original formulation of the geometric approach to supergravity in superspace and, in particular,of its application to the study of the Bianchi identities in superspace we refer the reader to [5,6] (see also [12]and Appendix A and B of [21]). Moreover, a concise review of the prescriptions on the supercurvatures inthe geometric approach to supergravity is also given in Appendix A of Ref. [22]. Here we are considering N = 1, D = 4. Spinor indices are denoted by α, β, . . . and in the sequel wewill frequently omit them to lighten the notation. The outer components of the curvatures are defined as those having at least one index along the ψ direction of superspace, while the components with indices only along the bosonic vierbein are called inner. Actually, this can be shownfrom both the study of the geometric Lagrangian and the sector-by-sector analysis of theBianchi “identities”. Within the latter approach, the Bianchi identities become relationsto be analyzed performing their split in the different sectors ψψψ , ψψV , ψV V , and V V V .This gives the expression of the outer components of the supercurvatures in terms of theinner ones, making the theory on superspace have the same physical content as the theoryon spacetime.Finally, we mention that since supersymmetry transformations are just Lie derivativesin superspace, they are easily derived from the (superspace) Lie derivative of the gaugefields using the formula in footnote 7, namely δ ǫ µ A = D ǫ A + ı ǫ R A , where D ǫ A is a gaugetransformation and the contraction is made with a supersymmetry parameter.We shall now apply the aforementioned prescription on the Bianchi identities to thecase of conformal supergravity with vanishing supertorsion. The superconformal algebra [23, 24] is generated by the set { J ab , P a , K a , D , A , Q α , Q β } .We introduce the 1-form fields { ω ab , V a , K a , D , A, ψ α , φ α } (see also [9, 10, 12]), respectivelydual to the vector fields generators of the superconformal algebra as given by (2.1) togetherwith A ( A ) = 1 , ψ α (Q β ) = δ αβ , φ α ( Q β ) = δ αβ . (4.2)The scale weight of the U(1) gauge 1-form field A , of the gravitino 1-form ψ , and of the conformino φ are, respectively,[ A ] = 0 , [ ψ ] = 12 , [ φ ] = − . (4.3) The relation between outer and inner comoponents of the supercurvatures is also referred to as the“rheonomy principle”. Actually this property is a consequence of the fact that the Lagrangian is constructedonly in terms of differential 4-forms in superspace, with the exclusion of the Hodge duality operator. R ab ≡ R ab − V [ a K b ] + ¯ ψγ ab φ ,T a ≡ D V a + D ∧ V a − i2 ¯ ψγ a ψ = ˆ D V a − i2 ¯ ψγ a ψ , T a ≡ D K a − D ∧ K a + i2 ¯ φγ a φ = ˆ D K a + i2 ¯ φγ a φ ,G ≡ d D + 2 V a K a − ψφ ,F ≡ dA + 2i ¯ ψγ φ ,ρ ≡ D ψ + 12 D ∧ ψ − Aγ ψ − i γ a φV a = ˆ D ψ − Aγ ψ − i γ a φV a = ∇ ψ − i γ a φV a ,σ ≡ D φ − D ∧ φ + 3i4 Aγ φ + i γ a ψK a = ˆ D φ + 3i4 Aγ φ + i γ a ψK a = ∇ φ + i γ a ψK a , (4.4)where ψ α and φ α are the gravitino and conformino 1-forms, dual to ordinary supersymmetryand conformal supersymmetry, respectively. We recall that D = d − ω is the Lorentzcovariant derivative, ˆ D is the Lorentz plus scale covariant derivative, and we have also takenthe opportunity to introduce, besides, a Lorentz plus scale plus U(1) covariant derivative ∇ . The matrices γ a , γ ab , and γ are the usual gamma matrices in four dimensions. Usefulformulas on gamma matrices can be found in Appendix A.The Bianchi identities obeyed by the supercurvatures (4.4) are D R ab + 4( T [ a K b ] − V [ a T b ] ) + ¯ φγ ab ρ + ¯ ψγ ab σ = 0 , ˆ D T a + R ab V b − GV a − i ¯ ψγ a ρ = 0 , ˆ DT a + R ab K b + GK a + i ¯ φγ a σ = 0 ,dG − T a K a + 2 V a T a − ¯ ψσ + ¯ φρ = 0 ,dF + 2i ¯ ψγ σ −
2i ¯ φγ ρ = 0 , ∇ ρ + 14 γ ab R ab ψ − Gψ + 3i4 F γ ψ + i γ a σV a − i γ a φT a = 0 , ∇ σ + 14 γ ab R ab φ + 12 Gφ − F γ φ − i γ a ρK a + i γ a ψ T a = 0 , (4.5)where ˆ D T a ≡ D T a + D ∧ T a , ˆ DT a ≡ DT a − D ∧ T a , ∇ ρ ≡ D ρ + 12 D ∧ ρ − Aγ ρ = dρ − γ ab R ab ρ + 12 D ∧ ρ − Aγ ρ , ∇ σ ≡ D σ − D ∧ σ + 3i4 Aγ σ = dσ − γ ab R ab σ − D ∧ σ + 3i4 Aγ σ . (4.6)21ne can now apply the prescription on the Bianchi identities to the present case, that iswriting the supercurvatures expansion as given in (4.1) and differentiating it to comparethe result with the Bianchi (4.5) expanded along the supervierbein basis. The closure ofthe resulting system of equations must occur sector-by-sector, that is along the ψψψ , ψψV , ψV V , and V V V sectors separately.Imposing vanishing supertorsion ( T a = 0) from the very beginning, a careful analysisshows that the superspace curvatures must have the following parametrization: R ab = R abcd V c V d + 2i ¯ ψγ c ρ ab V c ,T a = 0 , T a = T abc V b V c + ¯ ψ (cid:18) − σ ab − i2 γ ⋆ σ ab + γ ( a γ m σ m | b ) (cid:19) V b ,G = G ab V a V b ,F = F ab V a V b ,ρ = ρ ab V a V b ,σ = σ ab V a V b + (cid:18) − i2 ⋆ F ab γ b + 12 F ab γ γ b (cid:19) ψV a , (4.7)where for any 0-form U ab = − U ba we have denoted the corresponding Hodge-dual as ⋆ U ab = 12 ǫ abcd U cd . (4.8)As previously observed, the supersymmetry transformation laws differ from the gaugetransformations when the curvatures exhibit at least a gravitino ψ in their parametrization.In particular, in the case at hand this happens for R ab , T a , and σ , which indeed have a ψ in their parametrization (for the explicit form of the supersymmetry transformations ofthe fields we refer the reader to [9, 12]).Let us recall here that the quantities R abcd , T abc , G ab , F ab , ρ ab , and σ ab appearing inthe parametrization (4.7) are the so-called supercovariant field-strengths and they differ, ingeneral, from the spacetime projections of the supercurvatures. Indeed, let us refer e.g. tothe Lorentz supercurvature. Taking the components of R ab along dx µ ∧ dx ν , namely R abµν = R abcd V cµ V dν + 2i ¯ ψ [ µ γ c ρ ab V cν ] , we see that the spacetime components R abµν differ from thecomponents along the purely bosonic supervierbein, R abcd V cµ V dν . The quantity R abcd V cµ V dν ≡ R abµν | (cov) = R abcd V cµ V dν + 2i ¯ ψ [ µ γ c ρ ab V cν ] is the supercovariant field-strength. The same The scale of the supercurvatures components along the 2-vierbein sector is[ R abcd , G ab , F ab ] = − , [ T abc ] = − , [ T abc ] = − , [ ρ ab ] = − , [ σ ab ] = − . When doing the explicit calculations one can immediately simplify the starting general Ansatz by exploitingscale weight arguments. T a and σ . Instead, as in the present case the parametrizationsof G , F , and ρ have just components along two vierbein , covariant and supercovariantcomponents on spacetime are identified, that is we have G µν = G ab V aµ V bν , F µν = F ab V aµ V bν ,and ρ µν = ρ ab V aµ V bν .Besides the given parametrizations one also obtains the following constraints: G ab = 12 ⋆ F ab , ˇ R [ ab ] = − G ab = − ⋆ F ab , ˇ R ( ab ) = 0 , R = 0 , (4.9) γ ab σ ab = 0 ,γ a (cid:0) σ ab − i γ ⋆ σ ab (cid:1) = 0 , (4.10)and γ [ a ρ bc ] = 0 −→ γ c ρ ab = − γ [ a ρ b ] c , (4.11)the latter implying γ a ρ ab = 0 ⇒ γ ab ρ ab = 0 ,ρ ab + i γ ⋆ ρ ab = 0 . (4.12)Notice that using the first equation of (4.10), after some algebraic manipulation, we getthat the second of (4.10) reduces to the trivial identity 0 = 0.Let us just give a brief summary of the main steps of the cumbersome calculationsto recover the above results. The parametrization of R ab , G , and F , together with theconstraints in (4.12) and the fact that σ does not have components along two ψ , can beobtained by analyzing the ψψψ sector of the Bianchi for R ab , G , and F together with the ψψV sector of the Bianchi for T a and ρ . Considering the ψψV sector of the Bianchi iden-tities for G and F together with the ψψψ sector of the Bianchi for σ , the ψV V sector ofthe Bianchi for ρ , the ψψV sector of the Bianchi for R ab , and the V V V sector of the super-torsion Bianchi, one finds (4.9) and the parametrization of σ . Finally, the parametrizationof T a and the equations in (4.10) can be obtained by analyzing the ψV V sector of theBianchi for G and F together with the ψψV sector of the Bianchi for σ .The above results are in perfect agreement with the ones of [9,10] and [12]. In particular,the constraints in (4.12) are the ones used in [9, 10] (together with T a = 0). Moreover,the constraint γ a ρ ab = 0 in (4.12), is the same constraint fixed in [12] by using Lagrangemultipliers in the Lagrangian in order to recover superconformal invariance of the theory.We conclude that the solution given by eqs. (4.7), (4.9), (4.10), and (4.12) gives exactly thesame results as in Refs. [9, 10] and [12]. There the constraints were required by physicalarguments in order to have consistently supersymmetry invariance, while here we haveshown that they are a mere consequence of the geometrical structure of the theory expressedby the closure of the Bianchi identities. 23bserve that the constraints derived from the Bianchi identities turn out to be necessaryfor their closure, in a way quite analogous to the requirement that in the absence of auxiliaryfields the closure of the supergravity Bianchi identities only holds when the equations ofmotion are satisfied. However, in conformal supergravity the parametrizations for thecurvatures and the constraints recovered so far do not imply the equations of motion.One could then be surprised that we need constraints to have closure, since, after all,Bianchi identities, when no equation of motion is needed, are true identities. The pointis that the Bianchi identities would be true identities if we analyzed them in the enlarged superconformal coset of the basis gauge fields ( V a , K a , ψ α , φ α ), the other gauge fields ω ab , D ,and A being the factorized 1-forms dual to the generators belonging to the fiber. However,we want to have a physical theory on the ordinary supercoset spanned only by ( V a , ψ α ),which is a cotangent submanifold of the enlarged superconformal coset. The geometricconstraints of theory are then interpreted as the requirement needed in order to have aconsistent projection from the superconformal coset into the ordinary superspace.The fact that the study of the Bianchi identities leads to the constraint of conformalsupergravity has been also inferred in [14] in the context of an off-shell formulation of N = 2 supergravity with tensor multiplets. Here we have further highlighted and clarifiedthe geometric origin and meaning of the superconformal constraints in the case of N = 1, D = 4 conformal supergravity with T a = 0, whose understanding is rather fundamental inview of a future analysis including a non-vanishing supertorsion in the theory.Let us also mention that there are no independent differentials in the K a and Q α directions (as can be also deduced looking at (4.4) and (4.5)) so that one can write, usingalso scale weight arguments, K a = K ab V b + ¯ ψκ a ,φ = φ a V a , (4.13)where the 0-forms K ab , κ a , and φ a are a tensor, a spinor vector, and another spinor vector,respectively. Recall that K ab coincides with the spacetime components of K a only when ψ →
0, but since we are now in superspace the whole spacetime components of K a is givenby the supercovariant part of K a (that is K ab in (4.13)) plus the component along ψ . Whenone formulates the Lagrangian for the theory, the above components of K a and φ can bedetermined by studying the field equations of the theory (and this could be particularlyuseful in our future study where we will try to include a non-vanishing supertorsion).On the other hand, the aforementioned components can be also obtained by expandingthe supercurvature definitions given in (4.4) and using the geometric parametrization (4.7).24or the conformino components φ a we get φ b = 23 γ a (cid:18) i ρ (0) | ab − γ ⋆ ρ (0) | ab (cid:19) , (4.14)where we have exploited γ a ρ ab = 0 from (4.12) and used the definition of ρ given in (4.4)taking its 2-vierbein sector (which is the only sector appearing into the parametrization of ρ in (4.7)), that is ρ ab = ρ (0) | ab + i γ [ a φ b ] , ρ (0) | ab ≡ ∇ [ a ψ b ] . (4.15)Eq. (4.14) coincides, up to normalization and conventions, with the expression for theconformino ( φ µ = φ b V bµ ) found in [9]. Finally, notice that using the other results on ρ ab given in (4.12), making some algebraic manipulation we also obtain ρ (0) | ab + i γ ⋆ ρ (0) | ab = 0 . (4.16)Thus, using (4.16) into (4.14), we are left with φ b = i γ a ρ (0) | ab . (4.17)Similar arguments can be applied to find the expression for K a = K aµ dx µ = K ab V b + ¯ ψκ a by looking at the definition of R ab in (4.4) and using the parametrization for R ab in (4.7).More precisely, defining R ab (0) ≡ R ab + ¯ ψγ ab φ , (4.18)in such a way that fermionic contributions are taken into account in a straightforward wayby means of R ab (0) , we get R abµν − R ab (0) | µν = − V [ a [ µ K b ] ν ] ⇒ K µν = K aµ V aν = 12 (cid:18) ˇ R (0) | µν − g µν R (0) (cid:19) −
12 ˇ R [ µν ] − i2 ¯ ψ λ γ ν ρ λµ = 12 (cid:18) ˇ R (0) | µν − g µν R (0) (cid:19) + 14 ⋆ F µν − i2 ¯ ψ λ γ ν ρ λµ . (4.19)The latter coincides, up to normalization and conventions, with the same expression foundin [9] for K µν .We have thus shown that at the supersymmetric level, setting the supertorsion to zero,all the other constraints necessary for superconformal invariance and implemented in [12] We note that since there are no components of ρ along the outer basis ( ψ, V ) and ( ψ, ψ ), we have ρ µν = ρ ab V aµ V bν and therefore we can identify the a, b indices with spacetime anholonomic indices relatedto each other by the four-dimensional vierbein. This observation explains the meaning of the subsequentequation ρ (0) | ab ≡ ∇ [ a ψ b ] in (4.15), which would be senseless if a, b were interpreted as superspace indicesalong V a V b since the 1-form ψ in superspace is independent of V a by definition. In this paper we have shown that, in contrast with the claim of Ref. [1], it is actually possi-ble to construct a gauge theory of the conformal group in four spacetime dimensions with anon-vanishing torsion component. In particular, we have allowed for a non-vanishing axialvector torsion and found a sufficient condition to write a gauge theory for the conformalgroup. In this setup, invariance under proper special conformal transformations (confor-mal boosts) implies a differential condition on the axial vector torsion that, upon furtherdifferentiation, leads to a Maxwell-like propagation equation in a curved background forthe aforementioned axial vector. In the limit in which ˜ t a is set to zero we recover theconformal theory of [1].We have then given some preliminary results regarding conformal N = 1, D = 4supergravity. In particular, we have shown that the constraints introduced in [12] in thegeometric approach by the use of Lagrange multipliers in order to recover superconformalinvariance of the theory can be, in fact, directly obtained in the same geometric approachfrom the study of the Bianchi identities, just assuming vanishing supertorsion.This paves the way for future investigations that will be devoted to deepening theanalysis on conformal supergravity (this work is currently in progress [4]). In particular,since we have seen that at the purely bosonic level there exist the possibility of introducinga non-vanishing completely antisymmetric torsion without spoiling conformal invariance,we argue that something similar may also occur in the superconformal case. As a furtherremark, from a first glance we can say that (some of) the supersymmetric constraintsarising from the requirement of superconformal invariance would certainly be differentfrom the ones obtained in the case in which the supertorsion is set to zero in order toget a superconformal theory, and this in particular might cause something unexpected tohappen.Finally, let us say that our findings could also prove useful in the development a possi-ble four-dimensional extension of the theories presented in [25, 26] in the context of mod-ified/alternative theories of gravity. A detailed study in this direction could also unveilsome peculiar features of non-Riemannian degrees of freedom, together with a clearer un-derstanding of the potential relations occurring at the dynamical level among them.26 cknowledgements We thank our friends and colleagues Laura Andrianopoli and Mario Trigiante for a criticalreading of the manuscript, useful suggestions, and interesting discussions. L.R. would liketo thank the Department of Applied Science and Technology of the Polytechnic Universityof Turin, and in particular Laura Andrianopoli and Francesco Raffa, for financial support.
A Useful formulas on gamma matrices
We are working with Majorana spinors, satisfying ¯ λ = λ T C , where C is the charge conju-gation matrix. • Symmetric gamma matrices : Cγ a , Cγ ab , Cγ γ ab . • Antisymmetric gamma matrices : C , Cγ , Cγ γ a . • Clifford algebra : { γ a , γ b } = 2 η ab , [ γ a , γ b ] = 2 γ ab , γ ≡ − i γ γ γ γ ,γ † = γ , γ γ † i γ = γ i ( i = 1 , , , γ † = γ ,ǫ abcd γ cd = 2i γ ab γ , γ ab γ = γ γ ab , γ a γ = − γ γ ,γ m γ ab γ m = 0 , γ ab γ m γ ab = 0 , γ ab γ cd γ ab = 4 γ cd , γ m γ a γ m = − γ a ,γ a γ a = 4 , γ b γ ab = − γ a , γ ab γ b = 3 γ a ,γ ab γ c = 2 γ [ a δ b ] c + i ǫ abcd γ γ d , γ c γ ab = − γ [ a δ b ] c + i ǫ abcd γ γ d ,γ ab γ cd = i ǫ abcd γ − δ [ a [ c γ b ] d ] − δ abcd . (A.1) • Useful Fierz identities for N = 1 (for the 1-form spinor ψ ): ψ ¯ ψ = 14 γ a ¯ ψγ a ψ − γ ab ¯ ψγ ab ψ ,γ a ψ ¯ ψγ a ψ = 0 ,γ ab ψ ¯ ψγ ab ψ = 0 . (A.2)Irreducible 3- ψ representations:Ξ a (12) ≡ ψ ¯ ψγ a ψ , Ξ ab (8) ≡ ψ ¯ ψγ ab ψ + γ [ a Ξ b ](12) . (A.3)They satisfy γ a Ξ a (12) = 0, γ a Ξ ab (8) = 0, and we further have γ ab ψ ¯ ψγ a ψ = − γ a ψ ¯ ψγ ab ψ = − γ γ a ψ ¯ ψγ ab γ ψ = Ξ (12) b . (A.4)27 Some spinor identities : ¯ ψξ = ( − pq ¯ ξψ , ¯ ψ ( S ) ξ = − ( − pq ¯ ξ ( S ) ψ , ¯ ψ ( AS ) ξ = ( − pq ¯ ξ ( AS ) ψ , (A.5)where ( S ) is a symmetric matrix, while ( AS ) is an antisymmetric one; ψ and ξ are,respectively, a generic p -form spinor and a generic q -form spinor. References [1] M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, “Gauge Theory of the Conformaland Superconformal Group,” Phys. Lett. B (1977), 304-308[2] S. W. MacDowell and F. Mansouri, “Unified Geometric Theory of Gravity and Su-pergravity,” Phys. Rev. Lett. (1977), 739 [erratum: Phys. Rev. Lett. (1977),1376][3] P. K. Townsend and P. van Nieuwenhuizen, “Geometrical Interpretation of ExtendedSupergravity,” Phys. Lett. B (1977), 439-442[4] R. D’Auria and L. Ravera, Work in progress on conformal supergravity with a non-vanishing (super)torsion.[5] L. Castellani, R. D’Auria and P. Fr´e, “Supergravity and superstrings: A Geometricperspective. Vol. 1,2” Published in Singapore: World Scientific.[6] R. D’Auria, “Geometric Supergravity,” Review article from the book “Tullio Regge:An Eclectic Genius - From Quantum Gravity to Computer Play” , World ScientificPublishing Co. Pte.Ltd (2019), [arXiv:2005.13593 [hep-th]].[7] S. Ferrara and B. Zumino, “Structure of Conformal Supergravity,” Nucl. Phys. B (1978), 301-326[8] S. Ferrara, M. T. Grisaru and P. van Nieuwenhuizen, “Poincar´e and Conformal Super-gravity Models With Closed Algebras,” Nucl. Phys. B (1978), 430-444[9] M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, “Properties of Conformal Su-pergravity,” Phys. Rev. D (1978), 3179[10] P. K. Townsend and P. van Nieuwenhuizen, “Simplifications of Conformal Supergrav-ity,” Phys. Rev. D (1979), 3166 2811] E. S. Fradkin and A. A. Tseytlin, “CONFORMAL SUPERGRAVITY,” Phys. Rept. (1985), 233-362[12] L. Castellani, P. Fr´e and P. van Nieuwenhuizen, “A Review of the Group ManifoldApproach and Its Application to Conformal Supergravity,” Annals Phys. (1981),398[13] S. Ferrara, A. Kehagias and D. L¨ust,“Aspects of Conformal Supergravity,” [arXiv:2001.04998 [hep-th]].[14] N. Cribiori and G. Dall’Agata, “On the off-shell formulation of N = 2 supergravitywith tensor multiplets,” JHEP (2018), 132 [arXiv:1803.08059 [hep-th]].[15] R.O. Barut, Proc. Symp. on de-Sitter and conformal groups, Univ. of Colorado (1970).[16] F. W. Hehl, J. D. McCrea, E. W. Mielke and Y. Ne’eman, “Metric affine gauge theoryof gravity: Field equations, Noether identities, world spinors, and breaking of dilationinvariance,” Phys. Rept. (1995), 1-171 [arXiv:gr-qc/9402012 [gr-qc]].[17] F. Gronwald and F. W. Hehl, “On the gauge aspects of gravity,” [arXiv:gr-qc/9602013[gr-qc]].[18] D. S. Klemm and L. Ravera, “Einstein manifolds with torsion and nonmetricity,” Phys.Rev. D (2020) no.4, 044011 [arXiv:1811.11458 [gr-qc]].[19] P. S. Howe and U. Lindstr¨om, “Superconformal geometries and local twistors,”[arXiv:2012.03282 [hep-th]].[20] A. Garcia, F. W. Hehl, C. Heinicke and A. Macias, “The Cotton tensor in Riemannianspace-times,” Class. Quant. Grav. (2004), 1099-1118 [arXiv:gr-qc/0309008 [gr-qc]].[21] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria, S. Ferrara, P. Fr´e and T. Ma-gri, “N=2 supergravity and N=2 superYang-Mills theory on general scalar manifolds:Symplectic covariance, gaugings and the momentum map,” J. Geom. Phys. (1997),111-189 [arXiv:hep-th/9605032 [hep-th]].[22] L. Andrianopoli and R. D’Auria, “N=1 and N=2 pure supergravities on a manifoldwith boundary,” JHEP (2014) 012, [arXiv:1405.2010 [hep-th]].[23] J. Wess and B. Zumino, “Supergauge Transformations in Four-Dimensions,” Nucl.Phys. B (1974), 39-50 2924] S. Ferrara, “Supergauge Transformations on the Six-Dimensional Hypercone,” Nucl.Phys. B (1974), 73-90[25] S. Klemm and L. Ravera, “An action principle for the Einstein–Weyl equations,” J.Geom. Phys.158