Superconformal anomalies from superconformal Chern-Simons polynomials

We consider the 4-dimensional $\mathcal{N}=1$ Lie superconformal algebra and search for completely"symmetric"(in the graded sense) 3-index invariant tensors. The solution we find is unique and we show that the corresponding invariant polynomial cubic in the generalized curvatures of superconformal gravity vanishes. Consequently, the associated Chern-Simons polynomial is a non-trivial anomaly cocycle. We explicitly compute this cocycle to all orders in the independent fields of superconformal gravity and establish that it is BRST equivalent to the so-called superconformal $a$-anomaly. We briefly discuss the possibility that the superconformal $c$-anomaly also admits a similar Chern-Simons formulation and the potential holographic, 5-dimensional, interpretation of our results.


Introduction and summary
Historically, anomalies were first discovered by means of perturbative computations [1], [2].The BRST formulation of gauge theories uncovered a cohomological, non-perturbative, interpretation of anomalies as cocycles of ghost number 1 of the BRST operator acting on the infinite-dimensional space of fields [3].Later on it was understood that the anomaly cocycles of both Yang-Mills and gravitational theories are simply related to (a natural extension of the) secondary Chern-Simons characteristic classes [4], [5].This beautiful topological understanding of YM and gravitational anomalies simplified enormously their computation in arbitrary dimensions and for general gauge groups.It was also fruitful in many applications to string theory [6] and holography [7], [8].
While the BRST cohomological interpretation of anomalies is universal, the link between BRST anomaly cocycles and Chern-Simons invariants is not.To date, neither Weyl anomalies nor supersymmetry anomalies have been associated with Chern-Simons polynomials.This is at least one reason why their computation and classification are both less comprehensive and more intricate compared to Yang-Mills and gravitational theories [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].In this article we extend to conformal and supersymmetric theories the connection between anomalies and secondary Chern-Simons invariants.Specifically, we will show that the generalized Chern-Simons invariant associated to the d = 4, N = 1 Lie superconformal algebra computes one of the two independent anomalies of 4-dimensional superconformal gravity.
To put this result in the appropriate context, let us review the connection between YM anomalies and Chern-Simons polynomials as uncovered by Stora and Zumino [4], [20], [21], [22].Their idea is to introduce on d-dimensional space-time M d a generalized connection A with values in the Lie algebra g of the gauge group: A is defined to be the sum of the gauge field A and its corresponding ghost c, (1.1) They also introduced a generalized BRST operator δ where s is the BRST operator and d the de Rham exterior differential acting on forms.A is a generalized form of total fermion number -defined to be the sum of ghost number and form degree -equal to +1. 1 δ increases the total fermion number by 1.It is essential to keep in mind that, unlike ordinary forms, generalized forms of total fermion number n greater than the space-time dimension d do not in general vanish.d and s are taken to anticommute with each other: hence the nilpotency of δ is equivalent to the nilpotency of the BRST operator s.The cohomology of δ on the space of generalized local forms is isomorphic to the cohomology of s modulo d on the space of local ordinary forms.Therefore anomalies are obtained by integrating δ-cocycles with total fermion number d + 1 on the space-time manifold M d .These are local functionals of ghost number 1.
Given the generalized connection A and the generalized BRST operator δ one can define the corresponding curvature which is a generalized form of total fermion number +2 with values in the adjoint representation of the gauge Lie algebra g.The generalized curvature satisfies the Bianchi identity by virtue of the nilpotency of δ.Therefore, when d is even, g-invariant polynomials P d 2 +1 (F ) of F of degree d 2 + 1 are δ-cocycles of total fermion number d + 2 δ P d 2 +1 (F ) = 0. (1.5) Because of the curvature definition (1.3), P d 2 +1 (F ) is also δ-exact 1 Generalized forms Ωn = p+q=n Ω (p) q of total fermion number n are the sum of ordinary forms Ω (p) q of different form degrees p and ghost numbers q, such that q + p = n.
where Q d+1 (A, F ) are the celebrated (generalized) Chern-Simons polynomials.They are (non-gauge invariant) generalized forms of total fermion degree d + 1.The dependence of Q d+1 (A, F ) on the generalized connection A and curvature F is just the same as the dependence of ordinary Chern-Simons polynomials in dimension d + 1 on the ordinary connection A and curvature F .However, as stressed above, generalized Chern-Simons polynomials Q d+1 (A, F ) do not in general vanish in d dimensions.
The relevance of Chern-Simons polynomials in ordinary form cohomology is the following: ordinary Chern-Simons forms Q d+1 (A, F ) are not in general closed and, as such, they do not define de Rham cohomology classes.However there are situations in which some ordinary curvature characteristic class P d 2 +1 (F ), "accidentally" vanishes on manifolds M n of dimension n ≥ d + 2: in that case the corresponding Q d+1 (A, F ) is closed and it defines a characteristic class on M n of form degree d + 1, which is called secondary for this reason.
Going back to the BRST cohomology, the central observation of Stora and Zumino was that, for YM (and gravitational [23]) gauge theories, the generalized curvature F is actually "horizontal", which means that its higher ghost number components vanish (1.7) It follows that as ordinary forms of degree d + 2 do vanish in dimension d.Hence, in the YM BRST context one finds oneself in the precise analogue of the situation in which ordinary secondary characteristic classes arise in ordinary form cohomology: the ("primary" ) characteristic class P d 2 +1 (F ) vanishes and hence the generalized secondary Q d+1 (A, F ) is δ-closed δ Q d+1 (A, F ) = 0. (1.9) By integrating Q d+1 (A, F ) on M d one obtaines an anomaly cocycle.For Yang-Mills and gravitational theories all anomaly cocycles can be obtained in this way [24].The basic novelty one encounters when considering either supersymmetry or conformal symmetries is that the generalized curvature F defined by the corresponding BRST transformations ceases to be horizontal.Characteristic classes P d 2 +1 (F ) are then not guaranteed to vanish and this potentally negates the relevance of the generalized Chern-Simons polynomial Q d+1 (A, F ) to anomalies.
Although the non-horizontality of the generalized F is a generic feature of both conformal and supersymmetry transformations, let us illustrate how it comes about in d = 4 , N = 1 conformal supergravity, the field theory we are going to explore in this paper [25], [26], [27], [28], [29].Conformal supergravity is a "pure gauge" theory: it has neither auxiliary nor "matter" fields.Its gauge fields are 1-forms with values in the appropriate bundles each one in correspondence with the generators of the su(2, 2|1) Lie superconformal algebra: When one attempts to define the analogue of the generalized connection (1.1) one faces a complication which is common to all theories which include gravity: the ghosts associated to translations P a are not valued in the frame bundle but are instead valued in the space-time tangent bundle This requires treating diffeomorphisms differently to the rest of the Lie superalgebra transformations.As we will explain in Section 2, this has a twofold effect [30], [31], [32], [33].First, one has to introduce a novel BRST operator ŝ, "equivariant" with respect to diffeomorphisms where L ξ denotes the Lie derivative action along the vector field ξ µ .The sign in (1.13) is chosen so that the (diffeomorphism) equivariant ŝ includes all transformations other than diffeomorphisms.Correspondingly, the generalized connection (1.1) is defined to be 3 In other words, the ghost number 1 component of the generalized connection along P a is taken to vanish.Let us observe that the (1-form) components of the generalized connection along the bosonic (fermionic) generators of the Lie superalgebra are respectively anti-commuting (commuting).Hence it is convenient to introduce and take (fermionic) bosonic generators T i to be (anti)commuting: in this way h is anticommuting.Supersymmetric theories require one more step: the definition of the generalized BRST operator (1.2) must be modified to include one third term [32] δ = ŝ + d − i γ . (1.16) i γ is the nilpotent operator which contracts an ordinary form along the commuting vector field γ µ bilinear in the supersymmetry ghost ζ α : 4 Nilpotency of the generalized δ is equivalent to that of the BRST operator s: 5 Given these ingredients, one proceeds to define the generalized curvatures associated to the su(2, 2|1) Lie superconformal algebra exactly as in (1.3) where is a commuting generalized form of total fermionic number +2 which satisfies the generalized Bianchi identity (1.4).However, unlike the YM and gravitational case, H does not turn out to be "horizontal": rather one finds that H is an ordinary 2-form of ghost number 0 and λ 0 is a (non-vanishing) 1-form with ghost number 1, with values in the Lie superconformal algebra su(2, 2|1).We will denote the components λ i 0 of λ 0 as following the same order of the generators as in eq.(1.11).
The emergence of a non-vanishing non-horizontal curvature component λ 0 is intimately tied with presence of the extra term i γ in the definition of the generalized BRST operator (1.16): this term encodes the effect of coupling supergravity to YM gauge fields.The BRST transformations of the ghost fields are -for both bosonic YM and conformal supergravity -"geometric": they are fixed by the structure constants of the underlying Lie (super)algebra and nilpotency is ensured by the (super)Jacobi identities of the corresponding Lie (super)algebras.In the bosonic YM and gravitational case, the BRST transformation rules for the ghosts also uniquely fix the familiar, "geometric" BRST transformations rules for the connections: those transformations rules are not deformable.In short, the BRST transformations of both ghosts and connections are, for both YM and gravity, completely dictated by the geometry of the underlying Lie algebra.This ceases to be true in the supersymmetric context: as we will explain in Section 2, nilpotency of the BRST operator on the 5 This is a consequence of the BRST transformation of the diffeomorphism ghost ξ µ and the relations, valid on forms, for ŝ, d, and iγ ghosts of conformal supergravity determines the transformations of the connections only up to 1-forms of ghost number 1 which are i γ -closed -precisely because of the presence of i γ in the definition of the generalized δ, eq.(1.16).These 1-forms of ghost number 1 are the λ 0 's which appear in eq.(1.23), which indeed do satisfy Eq. (1.25) restricts the general form of the λ 0 's to be where X has ghost number 0 and is valued in su(2, 2|1).X is fixed by the requirement of nilpotency of BRST transformations on connections themselves, as we will explain in Sections 2 and 3: it turns out to be non-vanishing.Nilpotency of BRST transformations on gauge fields has one more implication: the ghost number 0 components H i of the generalized curvatures must satisfy certain constraints, which we will also review in Section 2. These are supersymmetric extensions of the familiar zerotorsion constraint of general relativity.Superconformal gravity constraints are algebraic equations for the gauge fields {ω ab , f a , ψα }, which can be solved to express them locally in terms of the physical fields {e a , b, a, ψ α }.It is an interesting fact that the non-horizontal components λ i 0 take values only in the "unphysical" directions {J ab , K a , S α } of the Lie superconformal algebra su(2, 2|1).
A priori, the lack of horizontality of the generalized curvature jeopardizes the Stora-Zumino mechanism to produce BRST anomaly cocycles.However horizontality of the generalized curvature is a sufficient but not necessary condition for the existence of secondary Chern-Simons classes.It is the vanishing of the characteristic classes P 3 (H) that is strictly necessary for the secondary classes to emerge.We therefore searched for su(2, 2|1) invariant cubic polynomials and found that there exists only one of them, up to a multiplicative constant.We computed the corresponding Chern class P 3 (H) and found, remarkably, that it indeed vanishes -despite the non-horizontality of H! The corresponding Chern Simons generalized form Q 5 (h, H) does therefore define, upon integration over space-time M 4 , a superconformal anomaly which we compute explicitly, in components and exactly to all orders in the number of fields, and present in Section 4, eqs.(4.13-4.16).The Chern-Simons anomaly cocycle Q 5 (h, H) is, by construction, invariant under rigid su(2, 2|1) transformations.It depends on all the ghosts (1.12) of the su(2, 2|1) Lie superalgebra, with the exception of the diffeomorphism ghosts ξ µ . 6In particular it depends on both the ghosts Ω ab and θ a associated, respectively, with local Lorentz and special conformal transformations.We will show that one can add a BRST-trivial cocycle to Q 5 (h, H) to obtain an equivalent super-anomaly cocycle Q equiv 5 (h, H) (eq.(5.50)) which is independent of both Ω ab and θ a .Such anomaly cocycle does not depend on the Weyl gauge connection b either: this is so since b and (a suitable completion of) θ a e a form a BRST trivial pair.It should be emphasized that the Ω ab and θ a independent cocycle is no longer invariant under the full rigid su(2, 2|1) Lie superconformal algebra.It leads to anomalous Ward identities which involve only the symmetric, conserved but not traceless stress-energy tensor T µν , the R-symmetry current J µ and the supersymmetry current S µ associated with the supersymmetry Q α .These are the (anomalous) Ward identities which are usually discussed in the literature.This allows us to identify the cocycle Q equiv 5 (h, H) with the so-called a-anomaly of superconformal gravity.Thanks to the "hidden" Chern-Simons origin this cocycle, we are able to write it in components, to all order in the number of both fermionic and bosonic fields, in eqs.(5.58-5.61).We believe such an explicit and complete expression has not appeared before.However it is important to observe that the Chern-Simons structure of the anomaly is only manifest in the superconformal invariant representative Q 5 (h, H) in which all the ghosts show up.
The rest of this paper is organized as follows: In Section 2 we review the BRST formulation of d = 4 , N = 1 superconformal gravity, which was first worked out in [31], by following a slightly different logic and formalism.This will allow us to describe the ingredients relevant to the computation of the anomaly.Our formalism will keep manifest the underlying covariance under the full Lie superalgebra su(2, 2|1) of the equations that determine the λ i 0 's.In this Section we also take the opportunity to elucidate why and how translations P a must be dealt with differently than other symmetries in the BRST context and why this entails, in the supersymmetric case, introducing the i γ term in the definition of the generalized BRST operator.
In Section 3, which also reproduces results already presented in [31], we describe how to solve the BRST nilpotency equations that both determine λ i 0 's and generate the constraints on the ordinary curvatures of superconformal gravity.Our presentation possibly clarifies why the solution to the BRST nilpotency conditions found in [31] is the only possible solution.We solve the constraints to express the fields {ω ab , f a , ψα } explicitly in terms of the physical fields {e a , b, a, ψ α }.
In Section 4 we describe the unique completely symmetric (in the graded sense) su(2, 2|1) invariant tensor and show that the corresponding generalized characteristic class P 3 (H) vanishes.To perform this latter computation we made use of FieldsX [34].We then present the ensuing secondary generalized Chern-Simons class which captures a superconformal anomaly.This is our main result.
In Section 5 we describe an anomaly cocycle equivalent to the Chern-Simons su(2, 2|1) cocycle, which is independent of Ω ab , θ a and b.It turns out that this is the "standard" non-superconformal invariant a-anomaly cocycle.We also write down the corresponding anomalous Ward identities for the currents T µν , J µ and S µ .
In Section 6 we briefly comment on the possibility that the c-anomaly also admits a Chern-Simons formulation.

BRST formulation of conformal supergravity
As mentioned in the Introduction, d = 4, N = 1 conformal supergravity is a "pure" gauge theory: all of its fields are 1-forms connections taking values in the appropriate bundles in correspondence to the generators of the su(2, 2|1) Lie superconformal algebra: The generators T i are graded:7 they satisfy (anti)commutation relations where f i k j are the structure constants of the d = 4 , N = 1 Lie superconformal algebra. 8he BRST formulation of conformal supergravity differs from that of pure (super) Yang-Mills theories in one crucial aspect.Let's delve a bit deeper into this distinction.
In (super)YM theories, one introduces in correspondence to each generator T i a ghost field g i with opposite statistics −(−1) |i| .The resulting Lie superalgebra valued combination is anti-commuting, and its BRST transformations are completely fixed by the structure constants of the Lie superalgebra: or, equivalently, where, as reviewed in appendix A, The (super)Jacobi identity [g, [g, g]] = 0 (2.8) ensures that the BRST rules (2.6) are nilpotent.Furthermore, the BRST transformations for the (anti-commuting) Lie superalgebra valued connection are also completely specified by the structure constants of the Lie superalgebra (2.10) For conformal supergravity -and for any theory which includes gravity -one has to proceed differently.In correspondence to diffeomorphisms one introduces an anti-commuting ghost ξ µ which is a vector field: there is no ghost valued in the P a sub-algebra.The BRST operator s acts on generic tensor fields ϕ via the Lie derivative L ξ 9 s ϕ = −L ξ ϕ + other gauge transformations, (2.11) and on the ghost ξ µ as follows γ µ is a quadratic function of the other ghosts whose precise form depends on the specifics of the gravitational theory one considers.We are going to exhibit its expression for superconformal gravity momentarily.Nilpotency of s requires that The way to deal with this situation is to disentagle translations from the other local symmetries.One introduces an "equivariant" (with respect to diffeomorphisms) BRST operator ŝ, whose action is defined on the smaller functional space of ghosts and connections which does not include ŝ involves only the ghosts g I corresponding to the gauge transformations other than translations.In the superconformal case these ghosts are Nilpotency of s is equivalent to the following relation for the equivariant BRST operator valid on the reduced field space which does not involve ξ µ .The action of ŝ on the ghosts g I cannot be simply defined by truncating the BRST transformation rule for the ghosts (2.6) to the g I : since the {T I }'s do not span a subalgebra, the truncated BRST transformations would not be in general nilpotent.Indeed, let {i} = {a, I} be the index running along the full Lie superalgebra and a the index running along the translations subalgebra: The Jacobi identity relevant for the nilpotency of (2.17) writes fJ where we introduced the ghost bilinear with values in the translations subalgebra We need therefore to introduce a suitable deformation of ŝ0 .One can start from the ansätz, dictated by ghost number conservation, which includes a term proportional to the gauge connection h I : where γ = γ µ ∂ µ is the ghost number 2 vector field which appears in the BRST transformations of the diffeomorphisms ghost (2.12) and i γ is the contraction of a form with the commuting vector field γ µ .Note that since γ µ is commuting.Moreover we must impose as consequence of (2.13).Therefore (2.24) We see therefore that we must take where λ I 0 are i γ -closed 1-forms which take value in the Lie superalgebra Eq. (2.25) fixes the vector field γ µ which appears in the BRST transformation (2.12) of the ghost ξ µ in terms of the structure constants of the Lie superalgebra: for su(2, 2|1) we obtain 10 (2.28) 10 The γ deformation is a signal of topological gravity [32] or supersymmetry [30], [33].Note that in the bosonic conformal case, fL a M = 0, because no commutator of TI generators TI gives P a (unlike the supersymmetric case, where [Q, Q] ∼ P ).Therefore, even if the truncation does not define an algebra, the truncated BRST operator is nilpotent and the γ deformation does not arise.
The condition (2.13) fixes the BRST rule for the connection e a , which is threfore "universal" for supergravity theories: In conclusion, the requirement of nilpotency of the BRST transformations on both the g I 's ghosts and the diffeomorphisms ghost ξ µ completely determines the BRST transformations of the ghosts, eqs.(2.12) and eqs.(2.21), which can be read off the structure constants of the gauge superalgebra.On the other hand nilpotency of the BRST transformations on ghosts determines BRST rules for the connections h I , eqs.(2.26),only up to i γ -closed 1-forms λ I 0 : We will see shortly that the λ I 0 are determined by the requirement of nilpotency of s on the connections h I : the λ I 0 's do not have an immediate interpretation in terms of the geometry of the gauge superalgebra.
We can now introduce the generalized-connection h i : where i runs along all the generators T i of the Lie superalgebra, with the understanding that the generalized connection along translations has no ghost number 1 component Moreover eqs.(2.21) and (2.26), dictate the form of the generalized BRST operator which differs from the Stora-Zumino analogue (1.2) for the i γ term, which encodes, in the BRST formalism, the "coupling" to supergravity.Generalized curvatures are defined in terms of the generalized differential δ and generalized connections in the usual way We can compute H by making use of eqs.(2.21) and (2.26) to obtain In other words the generalized-curvatures fail to be "horizontal" because of the λ I 0 which were left undetermined by the condition of nilpotency of the BRST operator on the ghosts. 11ne must therefore investigate the restrictions on the λ I 0 's coming from nilpotency of BRST transformations on the generalized connections: where H i are ordinary 2-form curvatures Equation (2.35) shows that not all the λ i 0 's can be taken to vanish, unless we impose H i = 0 for all curvatures, which would eliminate all propagating degrees of freedom from the theory.
As we made clear, eq.(2.35) is quite general: it is valid for any gravitational theory based on a Lie superalgebra with generators {T i }.A solution of this equation for the d = 4, N = 1 superconformal algebra su(2, 2|1) was found in [31].This solutions for the λ i 0 's also requires a set of constraints on the ordinary (both bosonic and fermionic) curvatures H i .We conducted with the help of FieldsX [34] a somewhat more systematic analysis of (2.35), which we summarize in Section 3 with the intent to ascertain if more general solutions exist.We recovered the same solution of [31] and nothing more.
Let us conclude this section by presenting the details of this solution.The BRST rules for the ghosts of su(2, 2|1) can be read off from (2.12) and (2.21): ŝ The BRST rules for the connections follow from (2.29) and (2.26):12 where the square brackets denote anti-symmetrization.
The explicit expressions for the two-form curvatures H i13 which include contributions from the full superconformal algebra, are: where D is the covariant derivative with respect to Lorentz, Weyl and U (1) R symmetries.
The non-vanishing λ I 0 's turn out to be: where we introduced the "modified" 2-form curvatures which have the property of transforming without derivatives of the supersymmetry ghost ζ under BRST transformations.Eqs.(2.35) which ensure the nilpotency of the generalized BRST operator δ on all fields, are satisfied by the λ i 0 's in (2.41a-2.41c)only on the subspace of fields defined by the set of constraints where R′ µν is the Ricci tensor constructed with the modified curvature R′ ab : R′ µν ≡ R′ µρ ab e b ρ e νa . (2.44) As we will review in the next section, these constraints can be solved algebraically to express the fields {ω ab , ψ, f a } as local functions of the independent fields {e a , ψ, a, b}.
3 Non-horizontal components of the curvatures and constraints This section, which can be skipped at a first reading and whose results reproduce those found in [31], is devoted to solving eqs.(2.35).In the generalized form approach, the failure of BRST nilpotency in the "big" field space of unconstrained generalized connections is the failure of the generalized Bianchi identity: where δ is defined in (2.32).Since the BRST rules of the ghosts are nilpotent in the "big" field space, the previous equation simplifies to or equivalenty, in components, Filtering in the ghost number, one gets: 1) the Bianchi identities for the ordinary curvatures (ghost number zero); 2) the BRST transformation rules for the ordinary curvatures (ghost number one); 3) s 2 on the gauge fields or equivalently the BRST transformation rules for the λ i 0 's (ghost number two), and 4) the i γ -closeness of the λ i 0 's (ghost number three): The equations at ghost number two are the same as eqs.(2.35).The trilinear Fierz identity for the commuting spinor together with eq.(3.4d), fixes the general structure of the λ i 0 's: , for the bosonic fields, (3.6a) for the fermionic fields. (3.6b) X i is a zero-form of ghost number 0. When λ i 0 is associated to a bosonic generator X i is Majorana spinor; when λ i 0 is associated to fermionic generator, X i is a matrix acting on the spinorial indices of ζ.
Eq. (3.4c) can be projected onto one component quadratic in the supersymmetry ghosts ζ and one component linear in ζ.Let us therefore correspondingly separate the part S in BRST operator ŝ which is proportional to ζ and associated to local supersymmetry [33]: The projection of eq.(3.4c) onto the component linear in ζ becomes This equation states simply that the λ i 0 's transform covariantly under all the transformations of the superconformal algebra other than the supersymmetry transformations.The projection of eq.(3.4c) onto the component quadratic in ζ, after taking into account eqs.(3.6a-3.6b),writes where we introduced the "modified" curvatures The dependence on the derivative of the ghost ζ in the BRST variation of H ′ i cancels between the first term and the BRST variation of ψ.Hence the modified curvatures H ′ i are supercovariant -i.e.their variations under (local) supersymmetry do not depend on derivatives of the supersymmetry ghosts -if we take X i proportional to the modified curvatures themselves.The possible modified curvatures involved in each X i are fixed by superconformal covariance (3.8). 14In particular the mass dimension of X i must be the same as that of h i increased by one half.We already determined the BRST rule of the vierbein in eq.(2.29), which implies that the corresponding λ 0 vanishes: Therefore T ′a = T a , and the nilpotency equation for the vierbein reads: λ Q 0 is necessarly proportional to the torsion, because the mass dimension of X ψ is 1 and the torsion is the unique curvature with the required mass dimension.The most general ansätz for λ Q 0 consistent with superconformal covariance is where By plugging (3.13) into the nilpotency equation (3.12), one obtains Thus, BRST nilpotency on e a requires both the vanishing of the torsion and of λ Q The torsion constraint (3.16) can be algebraically solved for the spin connection, expressing it in terms of e a , b and ψ: The torsion constraint is necessary to ensure BRST nilpotency in any supergravity theory, independently of any equations of motion.Note that bosonic connections of conformal supergravity have 48 off-shell degrees of freedom, while the fermionic connections have 24.Since the spin connection has precisely 24 components, the torsion constraint (3.16) ensures the matching between bosonic and fermionic degrees of freedom.
Let us turn to the λ 0 's associated to the Lorentz, Weyl and R-symmetry generators.Superconformal covariance dictates their form to be Nilpotency of the BRST operator implies that the BRST variation of a constraint is a linear combination of constraints.Hence By plugging the ansätz for λ J 0 and λ W 0 into this equation, one obtains: This is equivalent to and Eq. (3.23) is consistent with ρ not identically vanishing only if A Majorana spinorial two-form ρ carries the following representation of the Lorentz group: where ).The fermionic constraint imposes 16 equations which put the 4 ⊕ 12 to zero. 15hese 16 equations can be solved to express the conformal gravitino ψ algebraically in terms of the other fields: Therefore matching fermionic and bosonic degrees of freedom requires that 16 bosonic offshell degrees of freedom also be eliminated: We will see momentarily that these composite degrees of freedom are the f a fields.Since eqs.(3.24) and (3.25) have determined the λ 0 associated to Lorentz transformations (3.20a) to be BRST nilpotency on the gravitino ψ is equivalent to where ρ ≡ 1 2 ρ bc e b e c .One can verify that this equation is indeed satisfied by using both the Fierz identity for ζ and the fermionic constraint (3.26) for ρ.

By taking the covariant derivative of the torsion we obtain
or in components16 R[mn We call this equation Bianchi constraint, because it is a modified algebraic Bianchi identity for Rab .This equation, together with the fermionic constraint, allows one to write the antisymmetric part of the Ricci tensor of Rab in terms of the Weyl and fermionic curvatures as or equivalently, in terms of the modified superconformal Ricci tensor, Superconformal covariance dictates the following form for λ S 0 : where x, y, z are constants and We did not include a term proportional to R′ [ab] Γ ab in ansätz (3.37) since this term is equivalent to the one proportional to F W ab thanks to eq. (3.36).Inserting (3.37) into the BRST nilpotency equations for a and b, one arrives at )  The Ricci constraint puts the 9 s ⊕ 1 s to zero and the 6 a equal to F W ab .The Bianchi constraint sets the 1 ′ s ⊕ 9 a to zero, beyond also putting the 6 a equal to the F W ab .The independent components of R′ mn,ab are hence captured by the Weyl tensor 10 s .The 16 independent equations (3.43) associated to the Ricci constraint can be solved algebraically for the 16 independent f µ a : In conclusion the superconformal Lorentz curvature R′ ab , upon constraints, describes the 10 s Weyl tensor degrees of freedom of the physical (non-superconformal) curvature R ab .The Ricci degrees of freedom of the physical (non-superconformal) Riemann tensor R µν ab , which sit in the 9 s ⊕ 1 s representation, are instead described by the symmetric part of f ab .The remaining independent (off-shell) bosonic curvatures are the physical (non-superconformal) curvature tensors F W mn and F R mn . 17Nilpotency of the BRST transformation for ω ab is now ensured thanks to the Bianchi and Weyl-chiral constraints, along with expressions (3.31) and (3.42) for λ J 0 and λ S 0 .The nilpotency equation for ψ involves the yet to be determined λ K 0 , for which superconformal invariance dictates the following ansätz: 17 Indeed, R ab mn, F W mn , F R mn , f ab have, before constraints, respectively, 36, 6, 6 and 16 components, for a total of 64 bosonic components.The Bianchi, Ricci and Weyl-chiral constraints impose (1 + 9 + 6) + (1 + 9) + 6 = 32 conditions.Of the 64 − 32 = 32 free components, 20 are the components of the physical Riemann tensor, 6 are the components of F R and 6 those of F W . with x constant.By plugging this expression into the nilpotency equation (3.47) one obtains The ζη terms cancel out thanks to the fermionic constraint.The remaining terms ζζ terms all vanish thanks to the identity Γ ab ρ′ ab = 0, (3.50) which descends from the solution (3.30) of the fermionic constraint, if one also takes x = 2, that is Finally, the nilpotency equation for f a 4 The Chern-Simons superconformal anomaly When the constraints are satisfied, the superconformal generalized curvatures satisfy the generalized Bianchi identities We can therefore construct generalized Chern classes of total fermion number 6 by considering cubic polynomials of the curvatures which are superconformal invariant.From its definition, dijk is a "completely symmetric" (in the graded sense) tensor dijk = (−) |i||j| djik , dijk = (−) |j||k| dikj .(4.4) Superconformal invariance of dijk , together with the generalized Bianchi identity (4.2), ensures that We searched for solutions of Eq. (4.5) with symmetry properties (4.4) and found a single solution, up to a multiplicative constant: Since the super-covariant generalized curvatures are not horizontal it is not "a priori" guaranteed that the BRST-invariant generalized polynomial P 3 (H i ) gives rise to a secondary generalized Chern-Simons class of fermion number 5, i.e. to an anomaly cocycle.Since the non-horizontal components of the generalized curvatures are 1-forms, P 3 (H i ) has, in principle, components of form degrees 4 and 3: However, it is easy to see that P = 0, due to the the specific form of the superconformal invariant (4.7) that we found, and the fact that only {λ K 0 , λ J 0 , λ S 0 } are non-vanishing.Hence It is quite remarkable that, by taking into account both the expressions for λ i 0 's (2.41a-2.41c)and the constraints on curvatures (2.43a)-(2.43d),this 4-form turns out to vanish The vanishing of P 3 triggers the Chern-Simons secondary class mechanism: the generalized Chern-Simons polynomial of total fermion number 5 is an anomaly cocycle The Chern-Simons polynomial is completely determined by the super-invariant tensor dijk and the structure constants f i jk according to the universal formula If we denote by Q 5,N the part of the Chern-Simons polynomial (4.13) of degree N in the number of generalized forms h and H, we obtain the following explicit expressions, to all orders in all the fermionic fields18

An equivalent anomaly cocycle
The superconformal invariant anomaly cocycle (4.13) depends on all the ghosts {g I } of the superconformal algebra.Therefore the corresponding anomalous Ward identites involve all the currents associated to superconformal algebra generators {T I }.It is physically important to understand if there is a choice of local counterterms of the effective action which renders non-anomalous a subalgebra of the superconformal gauge symmetry.In the BRST formalism this is the same as asking if there exists a representative of the the same BRST cohomology class of (4.13) which is independent of a subset of the ghosts {g I }.
The superconformal invariant anomalous cocycle (4.13) does not depend on the diffeomorphism ghost ξ µ : As mentioned in the introduction this reflects the fact that there are no diffeomorphisms anomalies in 4-dimensions.General arguments suggest that, for the same reason, it should be possibile to choose an equivalent cocycle which does not contain the Lorentz ghosts Ω ab [23].In this section we will prove that one can indeed choose a representative in the same δ-cohomology class as (4.13) which is independent of the Lorentz generalized connection ω ab , and we will explicitly compute this anomaly cocycle.The anomalous Ward identities associated with this representative will be covariant under diffeomorphisms and local Lorentz transformations: they will therefore involve a stress-energy tensor which is both conserved and symmetric.In the following, we will refer to such a representative as the "Lorentz-equivariant" cocycle.
In this section we will also show that we can further choose a representative in the same BRST class as (4.13) which, beyond being Lorentz-equivariant, is also independent of the ghost θ a associated to special conformal transformations.This will involve a redefinition of the conformal supersymmetry ghost η.The resulting anomaly cocycle will also be independent of the Weyl gauge connection b.
In conclusion there is a anomaly cocycle equivalent to (4.13) independent of Ω ab and θ a which describes an effective action which is invariant under diffeomorphisms, local Lorentz transformations and local special conformal transformations.The anomalous Ward identities associated to this cocycle encode the non-conservation of the R-symmetry current J µ and of the supersymmetry current S µ , together with non-vanishing of both the trace of the conserved stress-energy tensor, T µ µ , and the trace of the supercurrent, Γ µ S µ .This is the form in which the anomalies of superconformal gravity are usually presented [35].It should be kept in mind however that this representative of the anomaly cocycle has lost the full superconformal rigid symmetry enjoyed by the Chern-Simons cocycle (4.13).

Removing the Lorentz anomaly
We want investigate if there exists a generalized form X 4 of fermion number 4 such that the cocycle Q5 (h, equivalent to the Chern-Simons superconformal invariant anomaly cocycle (4.13), does not depend on the generalized Lorentz connection ω ab .One expects such a representative to exist because it is generally understood that Lorentz anomalies are equivalent to diffeomorphism anomalies: since there are no diffeomorphism anomalies in 4-dimensions the Lorentz anomaly should be removable [23].However we are not aware of a constructive proof of existence of such a cocycle in the general superconformal context we are considering.Hence in the following we describe how to explicitly construct the Lorentz equivariant anomaly cocycle.
It is useful to introduce a set of commuting and constant ghosts κ ab of fermion number +2 and the "topological" nilpotent operator ∂ ω which shifts ω ab The action of ∂ ω on all other fields is taken to be trivial.The anti-commutator of δ and ∂ ω is (minus) a (rigid) Lorentz transformation δ Lorentz κ with commuting parameter κ ab : A Lorentz equivariant representative Q5 of the To solve (5.4) it is convenient to introduce a filtration for δ on the space of polynomials in h and H. Let be the total degree of a monomial h N h H N H .We can then decompose δ as the sum of δ 0 which commutes with N while δ 1 increases N by 1 Let us also define the operator i 0 , which commutes with It is immediate to verify that Both δ 0 and i 0 are nilpotent (5.12) The operator l 1 l 1 ≡ {δ 1 , i 0 } (5.13) increases the number of fields N by 1.It acts trivially on connections Any polynomial Q 5 of total fermion number 5 can therefore be decomposed in the sum of polynomials Q 5;N of fixed degree N : and one can move, by adding δ 0 -exact terms, the δ 0 from the H ′ = δ 0 h ′ to hit the Lorentz connection and produce Rab .Hence one can add to Q 5;3 a δ 0 -trivial term which eliminates the ω ab dependence.Explicitly, by choosing one produces a trilinear δ 0 -cocycle Q5,3 equivalent to the superconformal invariant which does not depend on the Lorentz generalized connection ω: The task is now to show that there exists a δ-closed extension of Q5,3 which is also Lorentzequivariant.One starts by considering the quartic extension of Q5,3 which satisfies the second of the equations (5.17): We can also introduce the bosonic operator ∂ R which shifts the Lorentz curvature by κ ab Since Q5;3 is Lorentz-invariant, δ 1 Q5;3 does not depend on ω: as the quartic extension of Q5;4 which is both equivalent to Q5;4 and independent of ω.The quintic extension is now also independent of ω.This was expected a priori: indeed, ∂ ω Q5;5 is both δ 0 -closed and i 0 -closed, and has N = 5.Hence, since Q5;5 contains no curvatures.Summarizing, the δ-cocycle is both equivalent to the superconformal invariant Q 5 (h, H) and Lorentz-equivariant.

Removing the special conformal anomaly
The BRST rules for the Weyl connection b µ are (5.44) If we define a new ghost θµ associated to special conformal transformations: the (b µ , θµ ) form, by construction, a trivial BRST doublet Relation (5.45) can be inverted to express the original ghost θ a in terms of the new θµ (5.47) The only ghost whose transformation rules contain θ is the special supersymmetry ghost η: (5.52) In the renormalization scheme in which the anomaly is described by the Lorentz-equivariant, θ-independent cocycle (5.50) we have which is equivalent to the Ward identities ) ) where the non-vanishing densities A W , A R , A Q and A S can be read-off eq.(5.50).
Let us introduce the combinations (5.57) We then have for the separate anomalies descending from the Lorentz-equivariant, θ a and b-independent anomaly cocycle (5.50): (5.61)

Comparison with a and c anomalies
In order to compare our Chern-Simons anomaly cocycle with the a and c anomalies of [15], we look at the bosonic part of the R-symmetry and Weyl anomalies, keeping the terms quadratic in the gauge fields: these terms capture anomalous Feynman diagrams with three external bosonic currents.We have for the R-anomaly (5.62)Note that the first term in this expression depends only on the Weyl tensor: the Ricci components of the Riemann tensors are encoded in the f a terms.By replacing the super conformal curvatures with the physical curvatures (and putting to zero the fermionic terms) all the f a dependence cancels out in (5.62) to give the result proportional to the so-called a-anomaly [35].Let us now turn to the bosonic part of the Weyl anomaly truncated to the terms quadratic in the gauge fields: the f a dependence cancels out to give in which the last term vanishes thanks to the constraint (3.53).Hence the Weyl anomaly is just proportional to the Euler-invariant, thereby confirming that our superconformal cocycle is equivalent to the a-anomaly.Summarizing, the dependence on the geometric Riemann tensor of the superconformal cocycle is contained both in the superconformal curvature Rab and the special conformal connection f a .Thanks to the constraints (the bosonic part of) Rab is essentially the Weyl tensor built with the geometric Riemann tensor R ab , while f a encodes the Ricci components of R ab .The superconformal Chern-Simons cocycle has the precise combinations of Rab and f a to produce the a-anomaly, thanks to the cancellation of the f a dependence in the cocycle which occurs after the replacements (5.63a).

Conclusions and open problems
We have remarked that the d = 4, N = 1 Lie superconformal algebra admits a single invariant completely symmetric (in the graded sense) tensor with 3 indices in the super-adjoint representation.We have also shown that the corresponding invariant polynomial, cubic in the generalized curvatures of superconformal gravity, vanishes -despite those generalized curvatures not being horizontal.Therefore the corresponding superconformal secondary Chern-Simons invariant is an anomaly cocycle.We computed this cocycle explicitly, in components and to all orders in the independent propagating fields of superconformal gravity.We showed that it is equivalent to the so-called a-anomaly of superconformal gravity, a superconformal extension of the Euler Weyl anomaly of bosonic gravity.Our result is best viewed as an extension of the Stora-Zumino paradigm for producing anomaly cocycles out of secondary Chern-Simons classes -generalizing it to the case, characteristic of supersymmetry and conformal invariance, in which generalized curvatures are not horizontal.
Superconformal gravity is believed to possess a second independent anomaly known as the c-anomaly, a superconformal extension of the Weyl anomaly of bosonic gravity, constructed from the Weyl tensor.Hence, it is natural to inquire whether the c-anomaly also lends itself to a Chern-Simons formulation.The fact that the d = 4, N = 1 Lie superconformal algebra admits a single 3-index completely symmetric (in the graded sense) invariant tensor -which we proved to correspond to the a-anomaly -would seem at first to rule this out.Nevertheless, as we have consistently highlighted in preceding sections, the superconformal curvatures with which one constructs invariant polynomials obey certain constraints.The cubic polynomial of the generalized curvatures that we considered in this paper is invariant even in the "big" field space of unconstrained curvatures and it vanishes only thanks to the constraints.It remains a possibility that cubic polynomials of the generalized curvatures exist which are δ-invariant only on the constraint surface.We leave to the future the task of searching for all Chern-Simons cocycles associated to cubic polynomials of the generalized curvatures which are δ-invariant up to constraints.
However, it's worth noting that polynomials which only involve the (supersymmetrization of the) Weyl tensor naturally arise in the superconformal formalism presented in this paper.As we have explained, the constraints restrict the (bosonic part of the) superconformal Lorentz curvature Rµν ab to coincide with the Weyl tensor.The remaining components of the physical Riemann curvature are captured, in our formalism, by the composite fields f a .In the cocycle that we presented, the Weyl anomaly receives two distinct contributions.
One is proportional to the Euler invariant constructed with Rµν ab .This contribution, by itself, would give an anomaly of c-type, since it depends only on the Weyl tensor part of the physical Riemann tensor.The superconformal invariant polynomial, however, also includes a second contribution to the Weyl anomaly, which involves the composite field f a .This term does depend on the Ricci part of the Riemann tensor.The two terms conspire together with the exact coefficients to produce the Euler invariant constructed with the full Riemann tensor -that is, the a-anomaly.It is clear however that alternative polynomials would typically include Weyl-tensor c-type contributions to the Weyl anomaly.The question that remains is whether any such polynomial, both δ-invariant and vanishing on the constraint surface, actually exists.
by the UKRI Frontier Research Grant, underwriting the ERC Advanced Grant "Generalized Symmetries in Quantum Field Theory and Quantum Gravity".
A d = 4, N = 1 Lie superconformal algebra In this appendix we review our conventions for the d = 4, N = 1 superconformal algebra.The bosonic and fermionic generators, the corresponding gauge fields and BRST ghosts are listed in Table 1.Note that that the physical fields e µ a , a µ , b µ and ψ µ α have canonical mass dimensions, 0, 1, 1, 1 2 respectively. 21Instead the composite fields f µ a , ω µ ab , ψµ α have non-canonical higher mass dimensions 2, 1, 3 2 .Table 2 lists the W and R charges of the fields of the theory, that we took into account in Section 3 to construct the possible forms for the λ i 0 's.Table 2. Weyl and R-charges of the connections. 21Restoring the gravitational constant, which we put to 1, graviton and gravitino would get the familiar mass dimensions, 1 and The coefficient A(R) in equations (B.12) is the anomaly coefficient: it describes the contribution to the superconformal anomaly of matter in a representation R of the superconformal algebra.

3 2 .
C(R) is the index of the representation, such that str R (T i T j ) = C(R)g ij (B.7) and g ij is the Lie superalgebra Cartan-Killing metric, defined as23   g ij = str super-adj (T i T j ).(B.8)f ijk are therefore related to the structure constants f i l j as follows:f ijk = f i l j g lk .(B.9)f ijk does not depend on the representation R because the Cartan-Killing metric is the unique rank-two (super)-symmetric invariant tensor for a given Lie-superalgebra.f ijk is completely "anti-symmetric" in the graded sense, that isf jik = −(−) |i||j| f ijk , f ikj = −(−) |k||j| f ijk .(B.10) d ijk (R) is instead completely "symmetric" in the graded sense d jik (R) = (−) |i||j| d ijk (R), d ikj (R) = (−) |k||j| d ijk (R).(B.11)The tensor d ijk (R) could in principle, for a generic superalgebra, depend on the representation.However we computed solutions to the invariance equation (B.6) for d = 4 , N = 1 superconformal algebra and we obtained two linearly independent solutions: one which coincides (up to a multiplicative constant) with the lowered structure constants f ijk and another solution with precisely the symmetry properties of d ijk (R).Hence there is a unique rank three invariant tensor of su(2, 2|1) with its symmetry properties, up to a multiplicative constant.Therefore,d ijk (R) = 2 A(R) d ijk , (B.12)where d ijk is independent of the representation R. d ijk is related to the tensor dijk which defines the invariant polynomial P 3 (H), (see eq. (4.3)), as followsstr R (H 3 ) = A(R) (−) |i||j|+|i||k|+|j||k| d ijk H i H j H k ≡ A(R) dijk H i H j H k = A(R) P 3 (H), (B.13) that is dijk = (−) |i||j|+|i||k|+|j||k| d ijk .(B.14)The sign factor relating d ijk to dijk , which is caused by the fact that both the curvatures and generators are graded, is invariant under exchange and cyclic permutation of its indices.Therefore does not change the symmetry properties (B.11).It is important to keep in mind that the "invariance" equation satisfied by the tensor dijk -which ensures BRST invariance of P 3 (H) -is different, although equivalent, from eq. (B.6) valid for d ijk : Ricci constraint.This equation again implies eq.(3.36)for the antisymmetric part of the Ricci tensor of R′ ab , but it also sets its symmetric part to zero.The tensor R′ mn,ab has 6 × 6 = 36 components.It transforms in the following representation of the Lorentz group: R′ mn,ab ∼ 10 s ⊕ 9 s ⊕ 1 s ⊕ 1 ′ s ⊕ 9 a ⊕ 6 a , (3.44) where the suffix s (a) denotes that the representation is symmmetric (anti-symmetric) with respect to the exchange of the two pairs of indices of R′ mn,ab . 1 s is the Ricci scalar R′ , 1 ′ s its dual ε mnab R′ mn,ab , 9 s ⊕ 1 s ⊕ 6 a is the Ricci tensor R′ ab and 10 s is the Weyl tensor representation.