Unfolding Conformal Geometry

Conformal geometry is studied using the unfolded formulation \`a la Vasiliev. Analyzing the first-order consistency of the unfolded equations, we identify the content of zero-forms as the spin-two off-shell Fradkin-Tseytlin module of $\mathfrak{so}(2,d)$. We sketch the nonlinear structure of the equations and explain how Weyl invariant densities, which Type-B Weyl anomaly consist of, could be systematically computed within the unfolded formulation. The unfolded equation for conformal geometry is also shown to be reduced to various on-shell gravitational systems by requiring additional algebraic constraints.


Introduction
Conformal geometry plays an important role in many areas of gravitational and high energy physics as well as in certain fields of geometry. It extends the diffeomorphism invariant Riemannian geometry with local rescaling symmtery, namely the Weyl (rescaling) symmetry. Conformal geometry can be also viewed as the geometry of the asymptotic boundary of the bulk spacetime with negative cosmological constant [1], and hence it can be used in the AdS/CFT correspondence [2]. Conformal gravity in four dimensions is an alternative gravitational theory enjoying Weyl symmetry besides diffeomorphism. Since its introduction by Weyl and Bach, many studies were devoted to it (see e.g. [3][4][5] and references therein). As a four-derivative gravitational theory in four dimensions, it is power-counting renormalizable as opposed to Einstein gravity and has many interesting features which might be relevant in phenomenological models of gravity: notably, it has the conformal symmetry which the early universe seems to exhibit, and hence conformal gravity or its variant may replace Einstein gravity in the very early time of the universe (see e.g. recent works [5,6] and references therein). More generally, when we consider various modifications of gravity, conformal geometry also plays a distinguished role since the diffeomorphism plus Weyl rescaling is the maximum gauge symmetry that a theory of symmetric rank-two tensor field can afford. Another prominent use of conformal geometry is in conformal field theories where it appears as Weyl anomaly (see e.g. [7] for an historical overview). Like the other anomalies, Weyl anomaly is subjected to the Wess-Zumino consistency condition, and the classification of Weyl anomalies by the relevant cohomological analysis has been innitiated in [8,9] with results up to dimension six. The structure in general dimensions, postulated already in [9], was confirmed first by using the technics of dimensional regularization on the effective gravitational action [10] and later by a cohomological analysis [11,12]. According to these results, a Weyl anomaly in d = 2n dimensions is the spacetime integral of a linear combination of the following densities multiplied by the Weyl rescaling parameter σ.
• Type-A anomaly, associated with a-coefficient: Euler density, E = a 1 ···a d R a 1 a 2 ∧· · ·∧ R a d−1 a d , where R ab is the Riemann curvature two-forms.
• Type-B anomaly, associated with c-coefficient: strictly Weyl invariant density, which is a specific contraction of (covariant derivatives of) Riemann tensors, including any full contraction of n Weyl tensors.
The explicit form or even the number of the non-trivial Weyl invariant densities -by "non-trivial Weyl invariant densities", we mean Weyl invariant densities which are not contractions of Weyl tensors -are not known in general dimensions but up to dimensions eight. In six dimensions, it was shown in [9,10,13] that there exists only one non-trivial Weyl invariant density (see also [2]). In eight dimensions, the authors of [14] showed, by employing purely algebraic methods based on a Weyl covariant tensor calculus [15] and with a help of computer program, that there exists five non-trivial Weyl invariant densities. The covariant derivative and the calculus used in [15] is closely related (see [16] for the relation) to the Thomas D operator and the tractor calculus for conformal geometry [17]. See e.g. [18] for more background in this topic. The current work originated from the attempt of understanding the above structure from a different angle. As Riemaniann geometry can be understood as the gauge theory of an isometry group, such as Poincaré or (Anti) de Sitter group, conformal geometry can be viewed as the gauge theory of conformal group SO (2, d). Such algebraic approaches to geometries have been pioneered by Cartan and Weyl, and known as Cartan geometry (see e.g. [19]). It incorporates not only Riemaniann, but also parabolic geometry of which conformal geometry is an example. In physics, there have been several attempts to obtain conformal gravity actions using so(2, d)-gauge fields. Analogously to the re-derivation of Einstein action using the isometry-algebra-valued gauge field [20] (see also [21]), four-dimensional conformal gravity action has been expressed as an action of so(2, 4)-gauge field subject to certain off-shell constraints [22][23][24] (see also the review [25]). In three dimensions, the Chern-Simons theory of so(2, 3)-gauge field provides a Weyl invariant theory with all necessary constraints integrated in it [26]. It has been also shown that conformal gravity actions can be obtained from an ambient space formulation [27] and from a dimensional reduction [28].
Yet another framework intimately related to the Cartan geometry is the Vasiliev's unfolded formulation [29][30][31][32][33]. It is basically the zero-form extension of free differential algebra introduced in the context of supergravity [34]. Thanks to the zero-forms, dynamical equations can be expressed in an integrable first-derivative form, and the consistency of dynamical system becomes purely algebraic. The unfolded formulation has been originally introduced for higher spin gravity [33,35,36] but can be applied to any dynamical system. In particular, a huge variety of free dynamical systems governed by the conformal algebra so(2, d) have been analyzed in the unfolded formulation [37,38], and it has been shown that the structure of the representations appearing there matches that of the Bernstein-Gelfand-Gelfand resolution (see e.g. [39]). In [40], Fefferman-Graham ambient construction has been integrated into the unfolded system as Hamiltonian constraints.
In this work, we unfold conformal geometry starting from the so(2, d) gauge connection. As mentioned above, the merit of the unfolding procedure lies in the zero-form sector and we closely look into this part and fill the details which are not delineated in the literature. By explicitly analyzing the first-order consistency of the unfolded equation, we find that the content of zero-forms corresponds to the spin-two off-shell Fradkin-Tseyltin module [37,41] (see also [42]), that is the module associated with the off-shell spin-two Fradkin-Tseytlin field [25]. In this paper, we are primarily interested in the unfolding of the off-shell system. Indeed, the unfolding formulation can be applied not only to dynamical systems but also systems without prescribed dynamics, namely, off-shell systems [38]. See [43,44] for recent works on the off-shell extension of higher spin gravity. An interesting point of off-shell unfolding is that it can be posteriorly reduced to an on-shell system by imposing additional algebraic constraints on the zero-form sector. In this way, once we unfold conformal geometry, we can reduce it to the on-shell conformal gravity, namely Bach-flat geometry, by eliminating some zero-forms with additional algebraic constraints. Moreover, by imposing additional constraints on both of the one-form and zero-form sectors, one can reduce it even to Einstein gravity or various modifications of it. If we restrict to the linear equation, then the unfolded off-shell spin-two Fradkin-Tseytlin system can be reduced to on-shell Fradkin-Tseytlin system, as well as partially massless or massive spin-two systems [45,46]. In this paper, we mostly consider the spin-two cases, but clearly, many of the above reductions can be generalized to higher spins. Let us note -to avoid a potential confusion of interpreting the above statements in the AdS/CFT context -that all these theories are defined in the same d dimensional spacetime.
Another merit of the unfolding of conformal geometry is that it allows to revisit the classification of Weyl anomalies, which was our original motivation. As we shall show below, the classification of Weyl anomaliesà la unfolding is essentially the same as the method proposed in [14]. The latter makes use of Weyl-covariant derivatives of Weyl tensors, which carry reducible Lorentz representations, to make an ansatz for Weyl invariant density. Instead, the unfolded equation is equipped with the zero-forms carrying irreducible Lorentz representations, and the ansatz for a Weyl invariant density is made by these zero-forms. Unfortunately, the unfolding does not gain sizable computational efficiency, and to revisit the eight dimensional classification or even to attack the ten dimensional problem, it would be inevitable to use a computer programming, like in [14], which was beyond the scope of the current work. Despite this limitation, we still find it useful in understanding the general structure of unfolding conformal geometry and the essence of the Weyl anomaly classification problem. The explicit knowledge of the zero-form content and their first-order gauge transformation allows us to do a few preliminary assessments of the classification.
The organization of paper is as follows. In Section 2, we begin with a review of the so(2, d) gauge formulation of conformal gravity and set the problem of unfolding conformal geometry, which can be approached perturbatively in the power of zero-forms. In Section 3, we explain how the linear part of the unfolded equation in the zero-form expansion defines a so(2, d) representation in the space of zero-form fields. This differs from the analysis of linearized equations around a specific background as the latter captures only the isometry part of so(2, d) . We solve the consistency conditions of the linear part by using cell operators and find explicit form of the equation up to linear order in the zero-form. In Section 4, we review the spin-two off-shell Fradkin-Tseyltin module, and show its so(2) ⊕ so(d) decomposition coincides with the content of the zero-form fields. We comment also on the subtle points on the active and passive actions and the role of dual representation. In Section 5, we sketch the structure of the unfolding at nonlinear orders and show how the nonlinear equations can be systematically determined by moving up from the zero-form field equation of the lowest conformal weight. In Section 6, we review the gauge symmetry of an unfolded system, and show how the non-trivial Weyl invariants could be determined as gauge invariant d-forms. In particular, we determine the unique quadratic part of the Weyl invariants in eight and ten dimensions. We also provide discussions on higher order parts. In Section 7, we show how various on-shell systems can be obtained by requiring certain algebraic constraints which are invariant under special conformal transformations. We also discuss how the off-shell conformal system itself can be viewed as a constrained system.

Unfolding conformal geometry
In this section, first we review the so(2, d) gauge formulation of conformal geometry, which has been known since [22][23][24] (see also [47] for a recent use of it). Then, we introduce the unfolding scheme to conformal geometry with a rather pedagogical account.

Gauge formulation of conformal geometry
Let us begin with setting the convention for the conformal algebra: the Lie algebra so(2, d) is generated by anti-Hermitian generatorsM AB with the Lie bracket, where η AB is the flat metric with signature (2, d) . Taking the basis with indices A = +, −, a and a = 0, 1, . . . , d−1 where η +− = 1 and η ab is the d-dimensional flat metric with signature From now on, all the Latin indices a, b, c, d, . . . are lowered and raised by η ab and η ab . Let us review now the gauge formulation of conformal geometry. We consider the gauge one-form taking value in so(2, d) algebra, where e a , ω ab , f a , b are one-form fields which are all independent at this stage. For geometric interpretation, we assume the one-form e a has components e a µ which are invertible, and the inverse is denoted by E µ a , which define vector fields E a = E µ a ∂ µ . The curvature two form, has the components, Here, D L is the Lorentz covariant differential, and R ab is given by the one-forms transform as 10) We would need a partial gauge fixing as well as imposing constraints to recover the usual geometry based on metric tensor. Let us review the set of constraints which bring the so(2, d)-gauge theory to conformal geometry. First, we impose the torsionless constraint, CP : which is modified by the presence of b. Here, we use the notation ! = to emphasize that it is a constraint that we decided to impose. We also impose the curvature for dilationD to vanish CD : From the Bianchi identity dF + [Â,F ] = 0, we find CP + BIP : BIĴ : where we implemented the constraints CP (2.14) and CD (2.15). Let us express (2.20) Then C ab,cd and C a,bc are given in terms of the trace of the differential Bianchi identity (2.18) requires C a,bc to be trace-free as well: In fact, the constraints CD and CĴ are not independent, and the former is a consequence of the latter together with other constraints. To recapitulate, we impose the following set of constraints, The above is the starting point of the unfolding machinery. Before moving to that, let us review how the usual conformal geometry can be recovered from this setting.

Reduction to metric formulation
All the constraints can be solved algebraically: • The constraint CP determines ω ab,c = i c ω ab (or ω ab = ω ab,c e c ) in terms of e a and b as • The constraint CĴ determines f (a,b) in terms of R ab = R a c ,bc : from (2.21), we find .
After solving all the constraints, the so(2, d)-gauge symmetry reduces to 2 We can fix the gauge symmetries associated withK a andĴ ab as follows.
• The gauge symmetry ofK a allows us to set b a to zero: Note that in this gauge, theK a symmetry must transformation under theD symmetry with the parameter, κ a = ∂ a σ . (2.38) • The gauge symmetry ofĴ ab is δ λ e a µ = λ ab e bµ , (2.39) which allows us to fix the degrees of freedom of e a µ besides those in The residual gauge symmetries are those ofD andP , • The gauge symmetry ofD gives the Weyl rescaling, δ σ g µν = 2 σ g µν . (2.41) • The gauge symmetry ofP gives the diffeomorphism, δ g µν = ∇ µ ν + ∇ ν µ , µ = e a µ a . (2.42) After the reduction, we find that R ab,cd and C ab,cd coincide with the usual Riemann and Weyl tensor, and P ab = f (a,b) and C a,bc = ∇ b P ac − ∇ c P ab with the Schouten and Cotton tensors. 2 In fact, the constraints are not invariant under the gauge transformation (2.13) with the parameter a : However, the gauge symmetries can be properly modified by a "non-geometrical" curvature term [23] so as to leave all the constraints invariant. In fact, this modification naturally arises in the unfolded formulation. See Section 6.1 for the details.

Unfolding conformal geometry
Let us now consider the unfolding of conformal geometry. Remind that we have used the equations, In the former set of equations, we slightly simplified the expressions by introducing K = and assigning the conformal dimensions ∆ = −1, 0, 0, 1 to e a , ω ab , b, f a , respectively. 4 Note here that the eigenvalue of the dilation operatorD is −∆. This reflects that the fields W [∆]ab··· carry in fact a dual (or contragredient) representation which is obtained by the anti-involution (P a ,Ĵ ab ,D,K a ) → (P a , −Ĵ ab , −D,K a ) . See Section 4.2 for more discussions on this point. In the following, we sketch the key reasoning of the unfolding scheme.
• The system (2.43) can be regarded as a set of equations for one-forms e a , ω ab , b and f a as well as zero-forms C ab,cd and C a,bc . Note that C ab,cd and C a,bc have conformal dimensions ∆ = 2 and 3, respectively. The zero-forms are completely, namely algebraically, determined by the equations, and hence no new degrees of freedom are introduced by employing them. About the one-forms, basically the equations tell how the (covariant) derivatives of the one-forms are determined by the other fields without any derivatives. Consequently, the one-forms are subject to certain conditions which are necessary for the system to be equivalent to conformal geometry.
• Viewing (2.43) as a dynamical system for the associated fields, that is, the one-forms e a , ω ab , b and f a and the zero-forms C ab,cd and C a,bc , it is more natural to introduce a new set of equations which determine the evolution-that is, the (covariant) derivatives-of C ab,cd and C a,bc : D K C ab,cd = (D L − 2 b)C ab,cd = e f C ab,cd,e + (pre-existing fields) , D K C a,bc = (D L − 3 b)C a,bc = e d C a,bc,d + (pre-existing fields) .
(2.46) 3 The maximal compact subalgebra of so(2, d) is not so(1, 1) ⊕ so(1, d − 1) but so(2) ⊕ so(d). However, the two subalgebras are intimately related as we shall comment later in Section 4.1. 4 Assigning the conformal dimension −1 to dx µ , the fields e a µ , ω ab µ , bµ, f a µ have conformal dimensions 0, 1, 1, 2 identical to the numbers of derivatives of the corresponding fields in the metric formulation.
On the right hand side of the equations, we have introduced a new set of zero-form fields C ab,cd,e and C a,bc,d besides what can be expressed in terms of pre-existing fields.
The new fields C ab,cd,e and C a,bc,d should be subject to a proper set of conditions so that the new equations (2.46) with the new fields neither introduce any new degrees of freedom nor remove any pre-existing degrees of freedom. For this, one need to examine the compatibility of the new equations (2.46) with the Bianchi identities (2.30).
• Viewing (2.43) and (2.46) as a dynamical system for the one-forms e a , ω ab , b and f a , and the zero-forms C ab,cd , C ab,cd,e , C a,bc and C a,bc,d , we can again introduce 'evolution equations' for C ab,cd,e and C a,bc,d in a similar manner as we did for C ab,cd and C a,bc .
• This procedure can be continued iteratively, and introduces infinitely many zero-form fields with infinitely many equations in a way that such an extension of the fields and equations does not alter the content of degrees of the freedom of the system.
In order to work with an infinite number of additional zero-form fields, we need to label them efficiently, and the subalgebra k = so(1, 1) ⊕ so(1, d − 1) can provide such a good label: 5 in the following any zero-form fields will be labeled as traceless fiberwise tensors with two groups of totally symmetric indices, subject to the Young projection condition, In this way, the fiberwise tensor carries an irrep under so(1, d − 1) corresponding to a two-row Young diagram. We adopt the following common short-hand notation, Sometimes, it will be more convenient to use what we will refer to as "depth" δ, than the conformal dimension ∆ : (2.50) The zero-form fields C [2]a(2),b (2) and C [3]a(2),b should be identified with the usual Weyl and Cotton tensors: If we relax the above identification condition, the zero-form equations that we will elaborate below has the capacity to describe a system of any integral spin.
The infinite amount of the K-covariant equations for zero-forms that we need to identify will take the following form, where the expansion coefficients E ]a(m),b(n)|c (C) is a highly non-trivial task, and hence we first identify the linear parts, which will determine the content of the zero-forms. The identification of non-linear terms can be worked out, in principle, order by order in ∆. Due to the boundness of ∆ ≥ 2, D K C [∆]a(m)b(n) will involve at most [(∆ + 1)/2] order terms.
3 First order unfolding of conformal geometry

First order unfolding
Let us consider the system only up to the linear order, where we have used the notation, 6 Remark that the linear terms of E [∆+1]a(m),b(n)|c (C) and F [∆−1]a(m),b(n)|c (C) are denoted by the actions ofP c andK c , respectively. It will become shortly clearly that they indeed correspond to the action of translation and special conformal transformation. Remind also that the action ofP a andK a on the space of zero-form fields is not yet defined. The equation (3.1) suggests to combine the linear terms with the K-covariant derivative as The Bianchi identity is the consistency of the equation (3.3) associated with Since the action of (D G ) 2 on C is at least quadratic in C, the following should hold.
and, we find that the consistency of the equation requires the operatorsP a andK a coincide with the actions of translation and special conformal transformation. Therefore, the linear part of D G can be viewed as a G = SO(2, d)-covariant derivative.
The most general form of aP a action on the space of tensors with two-row Young symmetry is simply where the operators P a 1± and P a 2± are the operators which map the tensors with the Young symmetry (m ∓ 1, n), and (m, n ∓ 1) to the tensor with the Young symmetry (m, n) . Since these operators are unique up to overall factors (see Section 3.3 for the details), the unknowns are only the proportionality constants which depend m, n and ∆. Similarly, the most general form of aK action can be written as with similarly defined operators K a 1± and K a 2± .

Linearization around (A)dS background
Before moving to solve the conditions (3.6), let us consider the linearization around a (A)dS background where the zero-forms all vanish: and the one-forms satisfyf a = Λē a ,b = 0 , (3.10) and hence, Repeating the analysis for the linear fluctuation, we find that the background G-covariant derivative reduces to a background H-covariant derivativē where H is the subgroup SO(1, d) for Λ > 0, SO(2, d − 1) for Λ < 0 and ISO(1, d − 1) for Λ = 0 generated byĴ ab andP a + ΛK a . In this case, the Bianchi identity gives the condition, Since the above, being identities, should not impose any relation between fields of different ∆, the three terms should separately vanish we can recover the three conditions among the ones in (3.6). When the cosmological constant vanishes, we have only one consistency condition, (P [cPd] C) [∆]a(m),b(n) = 0, which determines only theP a action, then theP a action alone defines the zero-form field content of linearized conformal geometry around flat spacetime (that is, the spin-two off-shell Fradkin-Tseyltin system). Since the field content of the linearized system should be the same as the field content of non-linear one, theP a action should be enough to determine the zero-form field content of conformal geometry. However, when we consider an on-shell reduction of the system, it is necessary to have the information of theK a action, which cannot be obtained from the linearization around flat space. It would be instructive to rewrite the linearized zero-form equation using the depth δ (2.50) instead of the conformal weight ∆: Remark that the half of terms in theP a ,K a action above preserve the depth, but not the conformal dimension, of the fields. If we impose the condition, then the system can rely on fields of a single depth: The resulting system is nothing but a non-conformal system such as massless, partiallymassless or even massive spin two. These systems are studied in [45,46]. The σ 1 ± and σ 2 ± operators used therein correspond in our case to Furthermore, if we impose the restriction, then we end up with only two operators σ − = σ 1 − =ē a P a 1− and σ + = σ 1 + = Λē a K a 1+ , which describe the massless spin-s dynamics. Therefore, the system (3.14) encompasses the dynamics of (partially-)massless and massive as well as conformal fields-the goal of the current paper-with a consistent choice of P a r± and K a r± . We will come back to this point later in Section 7.

Cell operators and Recurrence relations
As mentioned earlier, the operators P a r± and K a r± are proportional to the cell operators which adds or removes one box with index a to a two-row Young diagram [45,46,48,49]. These operators are unique up to proportionality and their precise expressions and properties are given in the latter references. Here, we use their realizations as differential operators acting on auxiliary variables: we contract the zero-form fiberwise tensors with two set of auxiliary variables u a and v a as Then, the irreducibility and traceless conditions of the tensors read 20) and the Lorentz generators act as the differential operator, The one-cell operators, denoted henceforth by Y a r± , can be defined as 7 where Π Y is the projection operator onto the space of traceless tensors of two-row Young diagram symmetry. Beside the one-cell operators, let us also introduce 'two-cell operators' defined by More explicitly, the one-cell operators, when acted on (m, n) tensors, have the form, We can express the product of two one-cell operators as two-cell operators as where m, n are the eigenvalues of u · ∂ u and v · ∂ v , that is, the length of the first and second rows of the Young diagram on which the operators act. Since the two-cell operators are independent, we can solve the Bianchi identities by expressing all the operators appearing there as linear combinations of the two-cell operators. In particular, the Lorentz generator can be expressed asĴ The operators P c 1± and K c r± are both proportional to Y c r± : where p [∆]m,n r± and k [∆]m,n r± are the proportionality constants. We will determine these constants by asking the operatorsP a andK a , defined by (3.7), (3.8) and (3.36), satisfy the conditions (3.6) which arose from the Bianchi identities of the zero-form equations and implies that the operatorsP a andK a form a representation of conformal algebra so(2, d) together withĴ ab andD . Note that P a r± and K a r± can be viewed as differential operators acting on C [∆](m,n) (u, v), whereasP a andK a cannot because they alter the conformal dimensions ∆.
Firstly, the condition [P a ,P b ] = 0 gives Secondly, the condition [K a ,P b ] = η abD −Ĵ ab gives two kinds of equations: homogeneous ones and inhomogeneous ones. The homogeneous equations are The inhomogeneous equations are When identifying consistent sets of equations for zero-form fields C [∆]m,n , one needs to take into account the ambiguities (or redundancies) of field redefinitions, This would affect the action ofP a andK a as (p (3.62)

Solutions
We can first consider the equations (3.37)-(3.42) from [P a ,P b ] = 0 , which are homogeneous equations for the p [∆]m,n r± coefficients. One can first observe that the 'seed' zero-form C [2](2,2) generate other zero-forms with integer ∆ ≥ 2 and their Lorentz label (m, n) are restricted since they should be related to (2, 2) by actions of ∆ − 2 one-cell operators. A simple reasoning reveals the following. For even and odd ∆, the admissible Lorentz labels of zero-forms are depicted as the black bullets in the (m, n) lattice of Figure 1 and In the following we shall label the conformal dimension in terms of the depth δ. The above content of zero-forms coincides with the basis tensors of Weyl invariants used in [15]. When we build theP a andK a action, we assumed that there exist one zero-form for each {δ}(m, n) label, which is now restricted to admissible ones. This 'multiplicity-one' assumption can be verified by demonstrating that the recurrence relations arising from [P a ,P b ] = 0 uniquely determines p {δ}m,n r± after fixing all field redefinition ambiguities. As mentioned before, the content of zero-form is determined solely by theP a action because if we linearize the system around the flat background, we only obtain the condition [P a ,P b ] = 0 . The other consistency conditions will be served to determine theK a action together with the value ∆.
Let us demonstrate how the recurrence relations   coefficients, namely theP a action, by making use of the freedom of field redefinition. This proves the correctness of the 'multiplicity-one' assumption. Let us restate the zero-form field content of the conformal geometry: with field redefinitions, we can set some of them to zero and solve the conditions (3.66) as 0 = 0. In this case, the fields whose redefinition could fix such p {δ}2,n 2± coefficients simply decouple from the theory, and hence we can remove them. Since we have less zero-form fields compared to the off-shell system, the resulting system will be an on-shell theory. See Appendix A for more details.
Next we can move to the inhomogeneous equations (3.50)-(3.53) arising from [K a ,P b ] = η abD −Ĵ ab . They uniquely determine the following quadratics: for n = 0, 1, 2 we obtain are not zero, we find that the coefficients k The decoupling of C [d] (2,0) implies that its dual state is a lowestD state and generates an invariant sub-representation, which can be quotiented out to get an "onshell" representation. In order to understand better the relation between the zero-form module and the highest weight representation of so(2, d), we review a few results of the representation theory in the following section.

Off-shell Fradkin-Tseytlin module
The actions ofP a andK a identified in the previous section, together with those ofĴ ab and D, define a so(2, d)-representation realized on the space of zero-forms C [∆]a(m),b(n) with field content (3.67). In this section, we demonstrate how one can recover the same field contents from an analysis of so(2, d) representations. For that, let us review a few basics: the (generalized) Verma module V(∆, Y) of so(2, d) is given by where [∆, Y] is the lowest-weight representation (or primary state), which carries a finite-dimensional irrep of so(2) ⊕ so(d). We denote the Young diagram Y as a row vector, (n, m, 1 k ) = (n, m, 1, . . . , 1 k ) . (4. 3) The Bernstein-Gelfand-Gelfand resolution provides various (non-unitary) representations of so(2, d) as a successive quotient of Verma modules V(∆, Y) [37,41] (see also [42]). The spin-s Fradkin-Tseytlin (FT) module, is the one related to the on-shell conformal spin-s field, but what we need is the module related to the off-shell conformal spin s, in particular the off-shell conformal spin two, namely conformal geometry. Above, D(1 − s, (s − 1)) is the module of the spin-s conformal Killing tensors. Note that the module associated with an on-shell system is the quotient of the module associated with its off-shell system by the module associated with the equation of motion. For the conformal geometry with s = 2, the equation of motion is the dderivative Bach equation. Therefore, conformal geometry must be associated with the module S(0, (2)) which does not involve any d dependent quotient. The logic extends to other spins, and S(2 − s, (s)) is the off-shell FT module, also sometimes referred to as the shadow module. Let us decompose S(2 − s, (s)) into so (2) The spin-two conformal Killing tensors are nothing but the so(2, d) adjoint representation, and hence we find The same result can be also obtained from S(0, (2)) = k=0 (−1) k V(2 + k, (2, 2, 1 k )) , (4.8) where the successive quotients represent the implementation of the Bianchi identity and the Bianchi identity of the Bianchi identity etc. The decomposition of the above into so(2) ⊕ so(d) modules are shown, in Appendix B, to reproduce the result (4.7). In (4.7), the lowest conformal weight state is (2, (2, 2)), which is the representation of the Weyl tensor, and the space coincides with the zero-form field content (3.67). Remark that in the unfolded equation, the zero-form fields carry finite-dimensional representations of K = SO(1, 1) × SO(1, d − 1), which is different from SO(2) × SO(d). However, they are related by the "Wick rotation" on the 0-th and d-th components, which maps the generatorsD =M 0 d to iD and (Ĵ ij ,Ĵ 0i ) to (Ĵ ij , iĴ 0i ). To repeat, the Wick rotation maps the Lie algebras so(1, 1) = Span R {iD} and so(1, d − 1) = Span R {iĴ ab } into the Lie algebras so(2) = Span R {D} and so(d) = Span R {iĴ ij ,Ĵ 0i }, respectively. It is important to note here that the Wick rotation does not alter the vector space on which the generators act, and the same vector space carries a representation of so(1, 1) ⊕ so(1, d − 1) as well as a representation of so(2) ⊕ so(d). In a vector space carrying an irreducible unitary representation of so(2) ⊕ so(d),D has a real eigenvalue ∆, and iĴ ij andĴ 0i are represented as finite-dimensional Hermitian matrices. The same vector space carries a finite but nonunitary representation of so(1, 1)⊕so(1, d−1), which was used to label the zero-form fields. Let us note also that the decomposition of a G = SO(2, d) representation into (infinitely many) non-unitary finite-dimensional representations of K does not imply that the Grepresentation is a non-unitary one. In fact, a state in the G-representation is not a finite but an infinite linear combination of finite-dimensional K-representations. If we adopt the standard norm of finite-dimensional representations (which is not positive definite since K is non-compact), the norm of a state in the G-representation may diverge. Therefore, we need to introduce a new scalar product for the G-representation, which will be singular for a finite-dimensional K-representation. This new scalar product can be positive definite, though the Fradkin-Tseytlin module is not of this type. What we discussed just above is analogous to what happens in the Taylor expansion of L 2 space: each Taylor coefficient has a divergent L 2 norm, whereas a L 2 state may have a divergent norm with respect to a well-defined scalar product in the space of Taylor coefficients.
Comparing (4.9) with (4.7), one can see that the two vector spaces are isomorphic as so(d) representations, but have different so(2) eigenvalues. One can obtain the same result (4.9) via the so(1, d) decomposition of D(d, (2)), which is the zero-form content of massless spin-2 field in its unfolded formulation. Here, one need to remind that the expansion in k should be considered as a Taylor expansion of a function even though it is not normalizable in Taylor basis but in a harmonic basis. Each of the (k + 2, 2) so(1,d) module can be further branched into so(d) irreps as The expansion (4.10) with (4.11) reproduces the decomposition (4.9) with m = k − n, but without the so(2) labels.

Zero-form module
In this section, we provide a more detailed explanation about the connection between the representations of zero-form fields and the off-shell FT module. All zero-form fields of conformal geometry can be packed into The basis vectorΨ a(m),b(m) can be realized various ways, for instance as we have introduced earlier in the paper, as a function of auxiliary variables: with ∆ = m − n + 2δ . Note here that we have introduce also a Lorentz scalar auxiliary variable t to generate fields of different conformal dimensions. The above realization is compatible with the grading, In terms of these, all the generators of so(2, d) will be realized as differential operators acting on the auxiliary variables: where P a r± and K a r± are defined in (3.36). Therefore, we constructed an oscillator representation which is realized on the space of functions , (4.19) subject to the conditions u · ∂ vS (0, (2)) = 0 and ∂ 2 uS (0, (2)) = 0 . It will be interesting to relate the above representation to the one that can be obtained from the reductive dual pair correspondence of (O(2, d), Sp(4, R)) (see [50] for a review of the correspondence).
Note thatS(0, (2)) has vectors with negative conformal weights, which are bounded from above (−∆ ≤ −2). The representationS(0, (2)) is in fact isomorphic to the dual representation (aka contragredient representation) of the off-shell FT model S(0, (2)). The isomorphism is given by the anti-involution, 8 ρ(P a ) =P a , ρ(K a ) =K a , ρ(Ĵ ab ) = −Ĵ ab , ρ(D) = −D . More precisely, we can define a non-degenerate bi-linear form ·|· for S(0, (2)) ×S(0, (2)) as Ψ  used in [37]. They are related by ρ = τ • ι through the involution ι given by where Π a(m),b(n)|c(m),d(n) is the projection operator onto the space (m, n), and Ψ  2)), respectively. Then, for any v ∈ S(0, (2)) and w ∈S(0, (2)) , the actions of an element X ∈ so(2, d) on each spaces satisfy v |X w = ρ(X) v | w . (4.24) The duality between S(0, (2)) andS(0, (2) should not be viewed as the coefficients, but as the dual basis vectors. Indeed, the active representation of the zero-form fields is dual to the passive representation, S(0, (2)). Therefore, the active representation of the zero-form fields is simply the off-shell FT module S(0, (2)). What is noteworthy in the active representation is that it is related to the gauge transformation δ C [∆]a(m),b(n) which is in fact nonlinear in C. We will come back to this point in Section 7.
5 Higher order unfolding of conformal geometry

General structure
Let us discuss the structure of the nonlinear unfolded equations for the zero-form fields. For that, let us label the zero-form fields by a collective index I = [∆]a(m), b(n) as In this notation, the nonlinear unfolded equations read simply 9 3) 9 One may think that it could be more useful to express the equation as then, act D G on it to get the Bianchi identity. However, such a computation could lead to an inconsistency: the LHS of the Bianchi identity is quadratic in C since (D G ) 2 = O(C), whereas the RHS is cubic in C. The mistake is due to the incorrect assumption of the G-covariance of the multi-linear forms hidden in the RHS of (5.2). More formally, the cocyle of the corresponding Chevalley-Eilenberg complex is covariant under K but not under G.
where the conformal weight of C I , E I and F I are ∆ I , ∆ I + 1 and ∆ I − 1 , respectively. Using the relations, [3]ab,c , we can simplify the action of D K on (5.3) as is what we have worked out in the previous section, and E I,a 1 and F I,a 1 are determined by the actions ofP a andK a . Moving to the second order, we find Since E I,a 1 and F I,a 1 are already determined, the above defines inhomogeneous linear equations for E I,a 2 and F I,a 2 . Finally, at the order n ≥ 3, we have which define again inhomogeneous linear equations for E I,a n and F I,a n where the coefficients are given by (E I,a m , F I,a m ) with m = 1, . . . , n − 1 . In this way, one can iteratively determine all power series coefficients E I,a n and F I,a n by solving the linear equations. The index I = [∆]a(p), b(q) takes infinitely many values, but at a fixed order n only finitely many ∆ can appear because the conformal dimensions ∆ is additive and bounded from below (∆ ≥ 2) : Moreover, for a finite ∆, finitely many (m, n) tensors appear. Therefore, the procedure of determining the functions E I,a (C) and F I,a (C) -that is, the unfolded equations for the zero-forms -can be decomposed into finite-dimensional linear equations. We have seen that the linear part of the unfolded equation for conformal geometry defines the representation S(0, (2)) of conformal group SO(2, d), but the system is not consistent without higher order completion. In general the unfolded equations can be viewed as a Lie algebroid with an infinite-dimensional base manifold corresponding to the zero-forms. In the conformal geometry case, the structure constant of the Lie algebroid is at most linear in C, whereas the anchor, given by E [∆],a (C) and F [∆],a (C), are higher order polynomials in C.

A few lowest ∆
Let us show more explicitly, but still schematically, how the nonlinear part of the consistency equations (5.8) and (5.9) can be organized as a series of finite-dimensional linear equations by arranging them in the total conformal dimensions ∆ tot = ∆ e + ∆ f + ∆ C .
• ∆ tot = 6 : In this case, we find two non-trivial conditions from e a ∧ e b and e a ∧ f b .
Even though that the above sets of equations are finite-dimensional linear equations, they are tedious to solve and hence it would be more desirable to use a computer program code, which is beyond the scope of the current paper.

Weyl invariant densities
In this section we discuss how Weyl invariants can be classified within the unfolded formulation of conformal geometry.

Gauge symmetry
Let us consider our system within the general framework of the unfolded formulation.
Our system has two kinds of differential forms, the one-forms taking values in the adjoint representation of so(2, d) and the zero-forms taking (infinitely many) values in the off-shell FT module S(0, (2)) of so(2, d), Their equations have the general structure, with the (generalized) Jacobi identity, The system is invariant under the gauge transformations, where w A are the (p − 1)-form gauge parameters associated with the p-form field W A . Applying this to our system, we find the "standard" gauge transformations for e a and b as δ e a = D K a + λ ab e b − σ e b , δ b = D K σ + a f a + κ a e a , (6.5) and "deformed" gauge transformations for ω ab and f a as The modification terms are proportional to the curvatures and corresponds to the "nongeometrical" terms considered in [23].
The zero-forms transform as involving nonlinear functions E I,a (C) and F I,a (C) . As we discussed earlier, the gauge transformations by λ ab and κ a can be used to reduce the system to the metric form, whereas the transformations generated by ε a and σ become the diffeomorphisms and Weyl transformations.

Constructing Weyl invariantsà la unfolding
Let us revisit the classification of Weyl invariants, which consist of the type-B Weyl anomalies, within the unfolded formulation of conformal geometry. Weyl invariants are the scalar densities made by curvatures, which are strictly invariant under Weyl rescaling, without relying on a total derivative term.
The Weyl invariants should correspond to a gauge invariant d-form within the unfolded formulation. The invariance underD andĴ ab requires the basis for the d-form to be made by e a and f a only. TheP a gauge symmetry is the diffeomorphism, so its invariance can be achieved only up to a total derivative term. On the contrary, theK a gauge symmetry is related to a Weyl rescaling, so we require the strict invariance without relying on an integration by part. From this, we can rule out the dependency of f a as it transforms with derivatives, which can never be compensated if integrations by part are not allowed. In the end, the ansatz for the Weyl invariants is The invariance under the Lorentz and dilatation is guaranteed by considering I d (C) where all the Lorentz indices of C are fully contracted without using any external tensors and the total conformal dimension is ∆ tot = d . The gauge variation under translation and special conformal transformations give With a total derivative term, the above can be expressed as + a 1 ···a d [d ε a 1 e a 2 ∧ · · · ∧ e a d ∧ e c + ε c e a 1 ∧ · · · ∧ e a d ] E I,c (C) ∂I d (C) ∂C I + a 1 ···a d (ε a 1 e a 2 ∧ · · · ∧ e a d ∧ f c + κ c e a 1 ∧ · · · ∧ e a d ) F I,c (C) ∂I d (C) ∂C I . (6.10) The first line is a total derivative term proportional to the translation gauge parameter ε a . The second line vanishes identically: due to the properties of antisymmetrizations. Only the third line poses as a non-trivial condition, which ensures the gauge invariance under translation and special conformal transformation at the same time. In the end, it is sufficient to ask the special conformal invariance of the density I d (C) : To find an explicit form of Weyl invariants, one needs to begin with a general ansatz I d (C) = I d (C [2] , C [3] , . . . , C [d−2] ) , (6.14) with a certain finite number of undetermined coefficients c i . For instance, for d = 4 we have only one term, whereas for d = 6, we have three terms, In the last line, we used the convention, By checking the special conformal transformations of I 4 and I 6 , we find I 4 is already conformally invariant: whereas the conformal invariance of I 6 requires The above defines the system of linear equations for c 1 , c 2 , c 3 , In any d, the above has one-dimensional solution space, where we fixed p [∆]2,1 2− = 1 using field redefinition freedoms (we will keep this choice in the following analysis). Likewise, once the form of F I,a (C) is determined, the problem of finding Weyl invariants becomes a pure algebraic exercise.

Quadratic Weyl invariants
As we have seen in 6d case, the ansatz for quadratic Weyl invariants are easy to handle so it allowed to determine the invariant using the explicit expression ofK a action. Let us extend this analysis to 8d and 10d. In these cases, the quadratic Lorentz scalars that are invariant under the linear part ofK a action should be complemented by cubic (and also quartic for d = 10) terms to compensate the nonlinear part ofK a action.
In the following, we determine the quadratic part of Weyl invariants, that is, the quadratic Lorentz scalars invariant under the linear part ofK a action. In 6d, there are 3 quadratic Lorentz scalars with ∆ = 6, and 2 quadratic Lorentz vectors with ∆ = 5, so a generic action ofK a would have left one dimensional solution space for quadratic Weyl invariant. This is to be contrasted with the situations in higher dimensions: as shown in Appendix C, there are 7 quadratic Lorentz scalars with ∆ = 8, and 8 quadratic Lorentz vectors with ∆ = 7; and there are 12 quadratic Lorentz scalars with ∆ = 10, and 19 quadratic Lorentz vectors with ∆ = 9. See Table 6.3 for the summary. Because of the "inversion of numbers" in 8d and 10d, we need explicit computations to identify the kernel ofK a action in these cases.
Eight dimensions As mentioned above, there are 7 quadratic Lorentz scalar with ∆ = 8 leading to the ansatz, The invariance of I 8 (C) underK a transformation requires With explicit values of k [∆]m,n r± , we find the solution space is one-dimensional:
TheK a invariance of I 10 (C) gives = c 12 Let us note that the densities I 8 and I 10 are invariant underK a action in any dimensions. Multiplied by the volume form, they become Weyl invariants in 8 and 10 dimensions.

Higher order Weyl invariants
In the previous section, we identified the quadratic part of Weyl invariants in d = 8 and 10, and this analysis can be straightforwardly extended to any even dimensions as we know the linear part of theK a action explicitly. In principle, to identify higher order parts, we would need explicit expressions of F I,a (C) up to ∆ I = d − 2, which requires many steps even for d = 8 as we have seen in Section 5.2. However, even without the nonlinear part ofK a action being identified, one can still guess the number of Weyl invariants. Let us explain this point in the following. TheK a gauge transformation is a linear differential operator on the space of C and hence can be expanded as powers of C as δ = δ (1) + δ (2) + δ (3) + · · · , (6.28) where δ (n) ∼ C n ∂ C . The commutativity ofK a transformation implies In 8d example, the Weyl invariants have at most cubic terms: and theK a invariance of the above is equivalent to The first equation is what we have solved in the previous section, and we already identified I (2) 8 . Turning to the second equation, we need to solve again linear equations for I (3) 8 with inhomogeneous term δ (2) I (2) 8 , which obscures the existence of the solution I (3) 8 . By taking the antisymmetrized δ (1) variation of the second equation, we find This tells that the inhomogeneous term δ (2) a I (2) 8 is δ (1) closed. If δ (2) a I (2) 8 is not in the δ (1) cohomology of the space of C 3 vectors with ∆ = 7, then we know the solution will exist, and it will be sufficient to identify the kernel of δ (1) in the space of C 3 scalars with ∆ = 8. We know from [14] this is indeed the case and the dimension of the kernel is four: the number of C 3 scalars and vectors are 11 and 7, and hence δ (1) is surjective.
Let us consider also the 10d example, where the Weyl invariants have at most quartic terms: I 10 = I (2) 10 + I (3) 10 + I (4) 10 . (6.33) TheK a invariance gives where the inhomogeneous terms in the second and third equations are δ (1) closed: Again, if the inhomogeneous terms are not in the δ (1) cohomology of the space of C 3 and C 4 vectors with ∆ = 9, then the solution will exist for I (3) 10 and I (4) 10 , respectively. Therefore, in such case, one can just identify the kernel of δ (1) in the space of C 3 and C 4 scalars with ∆ = 10.
Let us discuss more about the underlying algebraic structure. At a fixed order C m , the variation δ (1) = δ (1) K defines the cochain complex (A m,• , δ (1) K ) where A m,n : n ∧n → S(0, (2)) m and n is the subalgebra of so(2, d) generated byK a . By introducing the codifferential δ (1) P from theP a action, the associated homotopy operator H is given by We find that the δ (1) K cohomology is trivial if ∆ = d − n as it lies in the kernel of H. Weyl invariants precisely concern the other cases with ∆ = d − n and we need more detalied analysis. For explicit computations of the δ (1) K cohomology, it will be useful to recast all higher-order forms of C as differential operators in auxiliary variables. If we combine different C m , we have A n = ⊕ d 2 m=0 A m,n : n ∧n ⊗ P d 2 (S(0, (2)) → R and the differential and co-differential δ (1) K and δ (1) P are deformed to the full gauge variations δK and δP which are nonlinear in C. This can be viewed as a deformation of Lie algebra cohomology to a Lie algebroid one.

Reduction by constraints
So far, we have considered the unfolded equation for conformal geometry, that is, the offshell system for conformal gravity. In this section, we discuss how an off-shell system can be reduced to various on-shell systems. The reduction can be achieved by imposing certain algebraic constraints Φ I on the fields, Φ I (e, ω, f, b, C) = 0 . (7.1) A constraint generates infinitely many consequent algebraic constraints through (successive) gauge variations, δ · · · δ Φ I (e, ω, f, b, C) = 0 , where each gauge variation δ makes use of different gauge parameters, and they are not nilpotent operators. If the constraints are Lorentz tensors, Lorentz transformation will not generate any consequent constraint. If the constraints can be decomposed into homogeneous ones under dilatation, dilatation will also leave the constraints invariant. On the contrary, translation always generates additional algebraic constraints at the level of the unfolded system, but they are related to the derivatives of original constraints. Therefore, we can disregard the constraints which can be obtained by aP a transformation. We will refer to this class of constraints as descendant constraint. In this way, we are left with an assessment of the effect of special conformal transformation on the constraints. Below, we consider two cases of non-descendant constraints.
• Primary constraint: 10 the constraints which are left invariant under dilatation and special conformal transformation. If we further restrict to the constraints which contain linear term in fields, we find only two possibilities because only C [2](2,2) and C [d](2,0) satisfy the primary field conditionK a C [2](2,2) = 0 andK a C [d](2,0) = 0, respectively. The former case corresponds to the conformally flat geometry, 3) The latter case corresponds to the Bach flat geometry, where we have included non-linear terms O(C 2 ), which is necessary in compensating . This nonlinear term in (7.4) can be removed by a nonlinear field redefinition of C [d] (2,0) . If we consider primary constraints which are at least quadratic in fields, then the Weyl invariant densities belong to this class. They are all scalars, but one could equally consider tensor analogues of these nonlinear constraints. These constraints generalize the concept of primary fields to a nonlinear level because theK a transformation acts nonlinearly on the field C and the constraints are in general nonlinear functions of C.
• Section constraint: the constraints which are neither primary nor generate new constraints underK a action. In this case, the variation necessarily constrains the gauge parameters and breaks a part of conformal symmetry. A simple example is the Einstein equation, Φ a (f, e) = f a − Λ e a , (7.5) whose gauge variation, dropping the local Lorentz part, is [3]ab,c = 0 . (7.6) The above results in the symmetry breaking, and so(2, d) will be reduced to either so(1, d), so(2, d − 1) or iso(1, d − 1) depending on the parameter Λ. Another example is where µ ab (C) is a rank-two tensor function of C. The simplest non-trivial case is µ ab (C) = 1 2 C [4]ab , and it leads to a four-derivative theory. Therefore, in the unfolded system of conformal geometry, the dynamical equations of various on-shell gravitational systems can be treated as algebraic constraints.
In order to obtain conformal geometry, we have used the constraints (2.28). These constraints can be also regarded as primary constraints imposed on an even larger system, where none of two form curvatures are constrained: [3]a;bc , F ab J = e c ∧ e d B [2]ab;cd , FD = e a ∧ e b B [2]ab . (7.9) Here, the zero-form fields B [∆]a 1 ···am;b 1 ···bn are reducible Lorentz tensors (no Young symmetry condition is imposed on them) 11 with two groups of fully antisymmetric indices, a 1 · · · a m and b 1 · · · b n . They are subject to the Bianchi identities,  [2]ab;cd . The projection of (7.11) to such components of B [2]ab;cd contains f term, and hence these fields are neither descendant nor primary. We may call these constraints "secondary" since their K a variations vanish upon imposing primary constraints. Finally, all the components of B [3]a;bc are descendants of the primary or the secondary constraints. Therefore, imposing all the secondary constraints would trivialize the system and imposing only the traceful part of the secondary constraints gives conformal geometry. To recapitulate, in this enlarged unfolded system, which is nothing but the off-shell so(2, d) gauge theory with invertible e a µ , conformal geometry is obtained by imposing all the primary and a part of secondary constraints.
The unfolded system asks us to work with the space of functions in the zero-form fields. The vector space that the zero-form fields take value in carry an infinite-dimensional representation, which can be viewed as the Hilbert space of the corresponding quantum system. Then, it is natural to interpret the space of functions in the zero-form fields as the Fock space of the corresponding quantum field theory. In this regard, it would be tempting to understand whether and how a suitable quantization of the unfolded system can actually associate the space of functions in the zero-forms with the Fock space. After such a quantization, the Fock space will be endowed with nonlinear actions of SO(2, d). It will be equally interesting to understand the physical meaning of this nonlinear actions and the associated representations in the context of conformal field theory.

A Unfolding free fields
As shown in Section 3, theP a action is determined by the coefficients p subject to the condition (3.66). We also showed that all these coefficients can be determined by field redefinitions leading to the off-shell system (conformal geometry) or some of them can be set to zero and solve the condition (3.66) as 0 = 0. In the latter cases, the corresponding fields decouple from the system. In the following, we provide a few solutions of this kind.