Conformal Higher-Spin Gravity: Linearized Spectrum = Symmetry Algebra

The linearized spectrum and the algebra of global symmetries of conformal higher-spin gravity decompose into infinitely many representations of the conformal algebra. Their characters involve divergent sums over spins. We propose a suitable regularization adapted to their evaluation and observe that their characters are actually equal. This result holds in the case of type-A and type-B (and their higher-depth generalizations) theories and confirms previous observations on a remarkable rearrangement of dynamical degrees of freedom in conformal higher-spin gravity after regularization.


Introduction
Whether Einstein's gravity can be extended to a theory with larger symmetries than the usual diffeomorphism is a challenging question and attempts at answering it have brought many interesting results not only to the physics of gravitation but also to mathematics. An old example is provided by the fruitful exchanges between Weyl's theory of gravity [1] and conformal geometry. Higher-spin gravity is a much younger example and its underlying geometry remains somewhat elusive. However, its deep connections with important topics of modern mathematics (such as deformation quantization, jet bundles, tractors, etc) might reserve some surprises in store.
What we shall study in this paper is the theory which combines the conformal and higher-spin extensions of gravity, namely conformal higher-spin (CHS) gravity. The study of CHS gauge fields was started by Fradkin and Tseytlin in [2], where the higher-spin generalization of the linearized conformal gravity action was obtained. Subsequently, the cubic interactions of CHS fields were analyzed by Fradkin and Linetsky in a frame-like formulation [3,4] (see also [5,6] for early works on the CHS superalgebra) where the quartic order analysis nevertheless becomes difficult, similarly to the Fradkin-Vasiliev construction of the massless higher-spin (MHS) interactions [7,8]. In three dimensions however, the full non-linear action could be obtained in the Chern-Simons formulation [9][10][11]. The unfolded formulation on which Vasiliev's equations are based can be also applied to free CHS gauge fields (see e.g. [12][13][14][15][16][17][18]).
In fact, one of the simplest ways to understand CHS gravity in even dimensions is viewing it as the logarithmically divergent part of the effective action of a conformal scalar field in the background of all higher-spin fields [19][20][21], which can be computed perturbatively in the weak field expansion. In the context of higher-spin holography, the CHS action can be again related to (the logarithmically divergent part of) the on-shell bulk action of the MHS gravity in one higher dimension with standard boundary conditions. The CHS field equations can also be obtained as the obstruction in the power series expansion (à la Fefferman-Graham) of the solutions to the bulk MHS field equations [22,23]. This holographic picture makes it clear that the global symmetry of the CHS theory matches that of MHS theory (see e.g. [21]). This scenario actually extends beyond the case of the conformal scalar field, the construction of CHS theories via the effective action ensures that there is a one-to-one correspondence between free conformal field theories (CFTs) and CHS theories, 1 while the higher-spin holographic duality ensures that there is a one-to-one correspondence between them and MHS theories. Accordingly, we will denote the zoo of CHS theories following the by-now standard nomenclature (type A, B, etc) for MHS theories. 2 The type-A, -B and -C higher-spin gravities on AdS d+1 are dual to the (singlet sector of) vector model CFT d of free scalars, spinors and d−2 2 -forms, respectively.
1 See e.g. [24,25], as well as [26,27] where the effective actions have been calculated in the world-line formulation. 2 In the sequel, the somewhat ambiguous terms "CHS theory" or "CHS gravity" will stand either for the specific example of type-A CHS gravity or, on the contrary, for generic CHS theories. The context should make it clear.
CHS gravity shows several interesting properties. In particular, it shares the main drawback and virtues of Weyl's gravity. On the one hand, it is a higher-derivative theory so it is non-unitary. On the other hand, it is an interacting theory with a massless spintwo field in its spectrum whose interactions with the other fields include the minimal coupling, so it is a theory of gravity. Moreover, its global symmetries include conformal symmetry thus it is scale-invariant at the classical level. Consequently, the absence of Weyl anomalies would ensure that CHS gravity is UV-finite (see e.g. [2] for the original arguments). In fact, an advantage of CHS gravity over its spin-two counterpart is that it might be anomaly-free, as suggested by some preliminary tests. From a quantum gravity perspective, this motivates the study of CHS gravity as an interesting toy model of a UVfinite (albeit non-unitary) theory of quantum gravity. The spectra of higher-spin gravity theories (CHS and MHS) involve infinite towers of gauge fields. The corresponding infinite number of contributions require some regularization which opens the possibility that the collective behaviour of such infinite collection of fields may be much softer than their individual behaviour. In fact, due to the presence of huge symmetries, such systems have been observed to exhibit remarkable cancellation properties in their scattering amplitudes, one-loop anomalies, partition functions, etc. From a higher-spin theory perspective, CHS gravity can be viewed as a theory of interacting massless and partially-massless fields of all depths and all spins. It has the nice feature of being perturbatively local and of admitting a relatively simple metric-like formulation. Moreover, it appears to possess similar features to MHS theory. In fact, some classical MHS gravity could possibly arise from classical CHS gravity by imposing suitable boundary conditions around an anti de Sitter (AdS) background, in the same way that Einstein's gravity is related to Weyl's gravity [28].
The linear equation for the conformal spin-s field is local in even d dimensions and has 2s + d − 4 derivatives. Generically, the corresponding kinetic operator is higher-derivative and can be factorized into the ones for massless and partially-massless fields of the same spin (around AdS d ) [29][30][31] 3 (see also [34] for related discussions). This factorization allows us to compute easily its Weyl anomaly and it was found that the a-anomaly coefficient vanishes [35,36]. The c-anomaly would also vanish assuming a factorized form of the equation in the presence of non-vanishing curvature [36]. Unfortunately, such a factorization does not actually happen in general, as was shown in [30]. As a consequence, to determine whether the c-anomaly vanishes or not, one needs the linear conformal spin-s equation in an arbitrary gravitational background, which has been the subject of several recent papers [37,38] (see also [39,40] for other strategies).
Scattering amplitudes of CHS gravity also exhibit surprising features: four-point scattering amplitudes of external scalar fields, Maxwell fields and Weyl gravitons mediated by the infinite tower of CHS fields vanish [41,42]. There exists yet another approach to CHS theory using the twistor formalism [43]. The twistor CHS theory has the same linear action as the conventional CHS, but it is not clear that the two CHS theories are the same at the full interacting level. Recently, scattering amplitudes of twistor CHS theory have been studied in [44,45] (see also [46] in the context of conformal gravity).
In this paper, we will focus on the spectrum and symmetry algebra of CHS gravity. More precisely, we will show that its total space of on-shell one-particle states in even dimension d carries the same (reducible) representation of the conformal algebra so (2, d) as its global symmetry algebra (spanning the adjoint representation of the higher-spin algebra), i.e.
On-shell CHS = Higher-spin Algebra . (1.1) Strictly speaking, we will sum the characters of the CHS fields making up the spectrum of CHS gravity and compare them to the character of the corresponding conformal higherspin algebra, which is isomorphic to the higher-spin algebra of the associated MHS gravity. In such derivation, it is crucial to use the identity where χ KT , χ CHS and χ (P)M designate the so(2, d) characters of a Killing tensor and its associated CHS and (partially-)massless field, respectively. Here, q and x = (x 1 , . . . , x d 2 ) are related to the temperature and the chemical potentials for the angular momenta in the usual way (see (2.11) in the next section). The identity (1.2) relates the characters of a given CHS gauge field on flat spacetime M d , of the associated partially-massless field in AdS d+1 as well as the Killing tensor of the latter. 4 This identity was first derived for the massless totally symmetric case in [54] (see also [35] for related discussion), then further explored in [55]. The generalized version of the identity to partially-massless as well as mixed-symmetry cases is derived in Appendix A. The idea of the proof of (1.1) is to perform the sum of (1.2) over all fields in the spectrum of CHS gravity. After the sum, the terms corresponding to the last two terms in (1.2) cancel each other because of parity properties of the corresponding characters. A possible physical interpretation of the result (1.1) might be that the dynamical degrees of freedom of CHS gravity reorganize into its asymptotic symmetries. In this sense, CHS gravity is somehow "topological". This interpretation resonates with similar observations on the scattering amplitudes [41,42] and the one-loop partition functions on various backgrounds [55,56] of type-A CHS gravity.
The paper is organized as follows. In Section 2, we start by reviewing the field theoretical definition of type-A CHS theory as spelled out in [20,21], then move on to the derivation of the property (1.1). In Section 3, we extend property (1.1) to a couple of classes of higher-depth CHS theories, whose spectrum are made up of the CHS fields associated to the partially-massless fields making the spectrum of the type-A ℓ [23] and type-B ℓ theories. We start by describing the field theoretical realization of the type-A ℓ CHS theory in Section 3.1 and move on to show that (1.1) holds for the type-A ℓ in Section 3.2 and for the type-B ℓ in Section 3.3. We conclude the paper in Section 4 by discussing our main result and commenting on its implication for the computation of the thermal partition function and the free energy on AdS background of this theory. In particular, we point out that turning on chemical potentials for the so(d) angular momenta provides an alternative regularization of the sum over the infinite tower of higher-spin fields, which we also compare to the previously used regularization in the literature. Finally, Appendix A contains the derivation of identity (1.2), together with a more detailed description of the various modules of importance and their field theoretical interpretations in AdS d+1 /CFT d . In Appendix B we detail the d = 2 case as a toy model, while Appendix C contains the branching rule of the on-shell (totally-symmetric) CHS field module.
2 Type-A Conformal Higher-Spin Gravity 2.1 Field theory of type-A conformal higher-spin gravity The free theory of the conformal spin-s field in even d-dimensional Minkowski 5 spacetime M d , is described by the local action where h s is a totally symmetric rank-s tensor and P s TT is the projector to transverse and traceless symmetric tensors of rank s. The differential operator s+ d−4 2 compensates the non-locality of P s TT so that the action is local and conformally invariant.
• The field h s , referred to as off-shell Fradkin-Tseytlin (FT) field is a symmetric rank-s field of conformal weight ∆ hs = 2 − s. It is also called as the shadow field. Due to the projector P s TT , the action has the gauge symmetry with ξ s−1 a rank-(s − 1) symmetric tensor, η the Minkowski metric and σ s−2 a rank-(s − 2) symmetric tensor. Here, we used the schematic notation where all the indices are implicit.
• The conformal Killing tensor is the set of parameters (ξ s−1 , σ s−2 ) satisfying the conformal Killing equation i.e. the gauge parameters leaving the FT field inert under (2.2).
• The equation of motion of the action (2.1) reads This equation is referred to as the spin-s Bach equation where equalities that only hold on-shell will be denoted by the weak equality symbol ≈ hereafter. An equivalence class (2.2) of fields h s obeying this equation will be referred to as on-shell Fradkin-Tseytlin field.
• The action (2.1) can be also rewritten, after integrating by part, as where C s,s = P s,s T ∂ s h s is the (generalized) Weyl tensor of the FT field (here P s,s T denotes the traceless projector onto the two-row Young diagram displayed in (2.6) below). It is a traceless tensor with the symmetry of a rectangular two-row Young diagram, The Weyl tensor C s,s is a primary field with conformal weight ∆ Cs,s = 2 and is invariant under the gauge transformations (2.2). In particular, for s = 2 it corresponds to the linearized Weyl tensor.
So far we have considered the free theory of conformal spin-s gauge fields. An interacting theory of CHS gauge fields can be constructed from the effective (also called "induced") action of a free scalar field [19][20][21]26] in a higher-spin background. Starting from the action of a free complex scalar field φ coupled to higher-spin sources h s via traceless conserved currents J s =φ ∂ s φ (whereφ denotes the complex conjugate of φ) : we obtain the effective action where Λ is the UV cut-off. The logarithmically divergent part W log of the effective action W Λ is a local and nonlinear functional of the shadow fields h s . Moreover it can be shown (see e.g. [20]) to reproduce the free action (2.1) at the quadratic order: 9) and contains also the interaction terms O(h 3 s ), which can be perturbatively calculatedsee for instance [20,21,38]. Therefore, W log can be regarded as an action of interacting FT fields, up to the introduction of a dimensionless coupling constant κ : S CHS = κ W log . Note that the s = 0 term in (2.9) corresponds to the conformal scalar with d − 4 derivatives. Hence, for d = 4 the scalar field becomes auxiliary and drops out from the spectrum of CHS.
According to the AdS/CFT correspondence, W log should be equal to its AdS counterpart, that is, the logarithmically divergent part of the on-shell AdS action in the limit where the location of the boundary is pushed to infinity. In this way, the type-A CHS theory is linked to the type-A MHS theory. For instance, the free action S FT [h s ] is related to the Fronsdal action in AdS d+1 . Schematically, we have where K is the boundary-to-bulk propagator of Fronsdal field Φ s and R is the distance from the center to the boundary of the regularized AdS space. The interacting theory of CHS fields enjoys a non-Abelian global symmetry generated by the conformal Killing tensors. From the effective action point of view, this CHS symmetry is nothing but the maximal symmetry of the free scalar field. Hence, it coincides with the type-A MHS symmetry in AdS d+1 .
Suppose that we are interested in the quantum properties of CHS theory such as the one-loop free energy. Then, this quantity is closely related to the so(2, d) character of the free CHS theory. The latter character can be viewed as a single-particle partition function on S 1 × S d−1 (the conformal boundary of thermal AdS d+1 ) where we turn on, besides the temperature β −1 , the chemical potentials Ω i corresponding to the angular momenta [57,58], upon the identification, In order to compute the character of the free CHS theory, we need to determine first the characters of the individual so(2, d)-modules relevant in CHS theory.

Relevant modules
Let us start by reviewing the relevant modules in the free CHS theory (see appendix F of [54]). As usual in a conformal field theory, a primary field (together with its descendants) is described 6 by a (generalized) Verma module V ∆; Y , which is induced from the so (2)  where s 1 , . . . , s r are either all integers or all half-integers, corresponding to the spin, and r is the rank of so(d). The irreducible module obtained as a quotient of the Verma module V ∆; Y by its maximal submodule will be denoted D ∆; Y .
Rac. We introduce first the so(2, d)-module called Rac of order-ℓ (or ℓ-lineton) describing the conformal scalar field with ℓ th power of the wave operator as kinetic operator. Off-shell, a scalar field φ of scaling dimension d−2ℓ 2 is described by the Verma module V( d−2ℓ 2 , 0) where 0 stands for the trivial so(d)-weight. On-shell, such a scalar field obeying to the polywave equation, is described by the irreducible 7 module .
(2.14) 6 Strictly speaking, a generalized Verma module is the algebraic dual of the infinite jet space at a point of a primary field [59]. 7 Notice that, strictly speaking, when d is even the module (2.14) is irreducible if only if ℓ < d 2 .
This module is unitarizable for ℓ = 1, in which case it is simply called "Rac" (or scalar singleton). The value ℓ = d−4 2 for the higher-order Rac module gives precisely the s = 0 part of the CHS theory spectrum. Notice that the scalar FT field is absent in d = 4 and is unitary only for d = 6, in which case it corresponds to the usual Rac singleton as ℓ = d−4 2 = 1. When d 8 however, the order of the scalar singleton is greater than 1 and therefore this field is non-unitary, as the on-shell FT fields.
Conserved current. Let us now introduce the module describing the conserved spin-s current J s : i.e. a totally symmetric, traceless and divergenceless rank-s tensor. This current corresponds to the module D s + d − 2; (s) defined as the quotient . (2.16) The module V s+d−2; (s) contains symmetric and traceless rank-s tensors with conformal weight ∆ = s + d − 2 whereas the module V s + d − 1; (s − 1) is isomorphic to the divergence of such tensors. As a consequence, modding out this submodule is equivalent to imposing the conservation law (2.15). From the AdS d+1 perspectives, the unitarizable module (2.16) corresponds to the Hilbert space of the massless spin-s field with Dirichlet boundary conditions, i.e. the normalizable solutions thereof.
Off-shell Fradkin-Tseytlin field. The off-shell FT (or shadow) field corresponds to the module S 2 − s; (s) whose field-theoretical realization is a totally symmetric rank-s tensor field h s quotiented by the gauge symmetries (2.2). The case s = 0 is somewhat degenerate: the off-shell scalar FT field is simply S(2; 0) = V(2; 0). The precise grouptheoretical description of the shadow field in terms of Verma modules remains somewhat elusive. Notice that this module is not a (quotient of) Verma module(s) but is rather related to the contragredient thereof. Indeed, classical field-theoretical terms it is the dual of the conserved current J s with respect to the inner product d d x h s J s in the path integral. Equivalently, in CFT terms it is the algebraic dual of the conserved current J s with respect to the identity two-point function h s J s ′ = δ ss ′ . 8 For d odd, the module S 2 − s; (s) for the off-shell FT field is irreducible and isomorphic to the module describing the spin-s Weyl tensor C s,s , where the quotient by the irreducible submodule D 3; (s, s, 1) implements the generalized Bianchi identities.
On-shell Fradkin-Tseytlin field. For d even, the module S 2 − s; (s) of the off-shell FT field is reducible, corresponding to the possibility of imposing the (local) Bach equation. Interestingly, the module (2.16) may be interpreted as the left-hand-side of (2.4), i.e. the Bach tensor. Accordingly, the module corresponding to the on-shell FT field is in fact given by the following quotient: where U 3; (s, s, 1) implements the identitiesà la Bianchi obeyed by the on-shell Weyl tensor. Notice that the degenerate case s = 0 is consistent with (2.18) in the sense that the on-shell scalar FT field, which is an order ℓ = d−4 2 scalar singleton, is (2.20) Conformal Killing tensors. The last module we shall introduce is the one corresponding to conformal Killing tensors, Here V 1 − s; (s − 1) corresponds to the (large) gauge parameters of FT field h s and U 2 − s; (s) has the interpretation of pure gauge shadow fields, thus quotienting by this submodule corresponds to imposing the conformal Killing equation (2.3). Notice that there is a one-to-one correspondence between Killing tensors on AdS d+1 and conformal Killing tensors on M d , so the finite-dimensional irreducible so(d, 2)-module D 1 − s; (s − 1) has clear bulk and boundary interpretations.

Character of the Fradkin-Tseytlin module
The characters of the modules presented in the previous section can be related to the characters of the Verma module, is the character of the subalgebra so(d) of so(2, d) and the function P d is given, both for even and odd d, by . (2.23) Let us point out one important property of the above character, which is simply a consequence of the behaviour of P d under q → q −1 .
In this paper, we aim to find the character of the total linearized spectrum of CHS theory. For that, we need first the character of the spin-s on-shell FT field -which had been obtained in [54] using the Bernstein-Gel'fand-Gel'fand (BGG) resolution of so(2, d) [47] -then sum over all the spins. In order to avoid the technicalities, we shall present only key steps of the derivation, but interested readers can find more details in Appendix A.
Let us begin with the character of the spin-s on-shell FT field. From the definition (2.18), we first find χ D(2;(s,s)) = χ S(2−s;(s)) − χ D(s+d−2;(s)) , (2.25) where the off-shell FT field character χ S(2−s;(s)) and the Bach tensor (or conserved current) character χ D(s+d−2;(s)) are given by which relates the spin-s on-shell FT field module χ D(2;(s,s)) to the character of the conformal Killing tensor module χ D(1−s;(s−1)) up to the characters of a few Verma modules. The relation (2.28) does not yet express the character χ D(2;(s,s)) in terms of Verma module characters χ V(∆;Y) alone due to the presence of χ D(1−s;(s−1)) on the right-hand-side. In principle, we could further work out to get rid of the latter module using another relation for the modules but it will turn out to be useful to do the opposite. In fact, the expression (2.28) naturally leads to an interesting and suggestive expression of the character of the full CHS theory. This is thanks to the special property that the Verma module part of (2.28) enjoys: Note that the above property is a simple consequence of (2.24) and (2.27). This leads to the relation [54] χ S(2−s;(s)) (q, x) = χ D(1−s;(s−1)) (q, x) + (−) d χ D(s+d−2;(s)) (q −1 , x) , (2.30) which is valid in any dimension d > 2 (even or odd). Finally for even d, the character of the spin-s on-shell FT field module coincides with that of the conformal Killing tensor module up to just two additional terms: as follows from (2.25) and (2.29). Interestingly, both of these terms are given by the characters of the conserved current module, but one is with q while the other is with q −1 .
The formula (2.31) is the instance of the identity (1.2) which is relevant for type-A CHS gravity. It applies to the degenerate case s = 0 as well, except that the first term of the right-hand-side is absent in this case.

Character of type-A on-shell conformal higher-spin gravity
We shall use the relation (2.31) to derive the character of the CHS theory linearized spectrum. The theory contains the FT fields of spin 1 to ∞ and the scalar field with a kinetic operator containing d − 4 derivatives. Focusing first on the FT fields, we consider The first term in the right-hand-side of the equality is nothing but the character of the adjoint module of the CHS symmetry algebra. Re-expressing the two series in the second line using the Flato-Fronsdal theorem [61][62][63]: The second line of the above formula vanishes because the character of the Rac singleton obeys the property Finally, we find that the character of all the on-shell fields in the free CHS theory coincides with that of the global symmetry of CHS theory: This result can be understood as the equality (1.1) for type-A CHS gravity. 9 Actually, both sides of (2.36) involve divergent series which require some regularization. However, the equality (2.36) itself only assumed the validity of the Flato-Fronsdal theorem which does not need any regularization. 10 Consequently, confident in the validity of (2.36) one might somehow reduce the issue of regularizing the character of the CHS spectrum (the lefthand-side) to the one of the higher-spin algebra 11 (the right-hand-side). By construction, the corresponding regularization of CHS theory would preserve higher-spin symmetries, an important requirement of a sensible regularization but which is usually not guaranteed. There is another virtuous corollary of the relation (1.1): the Casimir energy of free CHS theory on the Einstein static universe R × S d−1 is ensured to vanish in any regularization consistent with (1.1). In fact, the vanishing of the Casimir energy is ensured when the partition function is invariant under the map q → 1/q (see [65]) and this property automatically holds for the right-hand-side of (1.1), since the character of each Killing tensor module obeys this property. Actually, the a-anomaly of CHS gravity is also guaranteed to vanish by virtue of this property of the character. Indeed, the a-anomaly of a d-dimensional FT field coincides with the difference of the free energy of the associated massless field in Euclidean AdS d+1 with Neumann boundary condition and the same with Dirichlet condition [35]. Since the free energy with Neumann boundary condition is simply minus that with Dirichlet condition (the contribution of Killing tensor module simply vanishes), the a-anomaly of the d-dimensional CHS gravity is just minus two times the free energy of MHS gravity in Euclidean AdS d+1 . The cancellation of the latter can be shown using the method of character integral representation of zeta function [66][67][68].

Character of type-A off-shell conformal higher-spin gravity
We can also derive an off-shell version of the identity (1.1). From (2.18), the spin-s off-shell FT field module is related to the on-shell one by χ S(2−s;(s)) = χ D(2;(s,s)) + χ D(s+d−2;(s)) , (2.39) whereas the off-shell FT scalar S(2; 0) = V(2; 0) is related to the on-shell one by Let us illustrate why (1.1) does not hold for minimal CHS gravity with only even spins in the spectrum.
In such case, the Flato-Fronsdal theorem involves a symmetric plethysm of the Rac module: Then (2.31) and the analogue of (2.35) imply the relation where the extra term on the right-hand-side has no clear interpretation in this context (as it decomposes into an alternating sum of the characters of massless AdS d+1 fields of all integer spin). 10 The only assumption is that it holds for each sum of characters (in q and in 1/q) separately, despite the fact that the corresponding power series have distinct region of convergence (|q| < 1 and |q| > 1). 11 Some convergence and regularization issues of the latter were addressed in [64].
where the first term on the right-hand-side is absent in d = 4. In fact, the above off-shell scalar becomes an auxiliary field in four dimensions. In odd dimension d, one can make use of (2.30). Summing over the characters of these off-shell field modules, we arrive at where the last term on the right-hand-side, the linearized spectrum of MHS gravity around AdS d+1 with Dirichlet boundary conditions, corresponds to the last series in (2.41). The two terms on the right-hand-side of (2.42) have natural interpretations in terms of the higher-spin algebra: they are important modules of the latter. The first term is the adjoint module while the second term is the so-called twisted adjoint module of the higher-spin algebra. From a holographic perspective, another interpretation of this last result is possible, purely in terms of bulk fields. For a spin-s massless bulk field, two boundary conditions are available: either the standard ("Dirichlet") boundary condition which allows normalizable bulk solutions corresponding to a conserved current J s with conformal weight ∆ + = s+d−2, or the exotic ("Neumann") boundary condition which allows non-normalizable bulk solutions corresponding to a shadow field h s with conformal weight ∆ − = 2 − s. Accordingly, the bulk theory with standard (respectively, exotic) boundary condition for all fields will be referred to as Dirichlet (respectively, Neumann) MHS theory. They are summarized and compared in Table 1. The holographic dual of Dirichlet MHS theory is a free scalar CFT [69,70]. Following the usual considerations on double-trace deformations and holographic degeneracy [71][72][73], applied for all spin-s conserved currents, one is lead to the conclusion that the holographic dual of Neumann MHS theory is CHS gravity. This scenario has been extensively discussed for the MHS theory around AdS 4 and Chern-Simons CHS theory around M 3 (see e.g. [35,74,75]), but the logic works for any dimension (see e.g. [50] for any spin at free level and [76] for spin-two at interacting level).

Bulk MHS
Bulk ∆ s Boundary Boundary theory field operator theory This exotic type of holographic dualities where both sides can be gravity theories (though of Einstein vs Weyl type) has been denoted "AdS/IGT" in [35] -where IGT stands for induced gauge theory -in order to distinguish it from standard AdS/CFT correspondence. This type of holographic duality is somewhat less familiar and is a subtle one, so let us expand a little bit and present some details. One starts by adding doubletrace deformations for all U (N )-singlet primary operators J s (with s = 0, 1, 2, · · · ) bilinear in the N complex scalar φ i : These deformations explicitly break all gauge symmetries of the background fields h s . Notice that essentially all 12 these deformations are irrelevant in the infrared (IR). The corresponding effective action is defined as The standard Hubbard-Stratonovich trick corresponds to the introduction of auxiliary fields σ s through Gaussian integrals as follows: where we neglected an infinite prefactor. Note that σ s does not have any gauge symmetries. The integration over the dynamical scalar field φ is now again a Gaussian integral which can now be performed leading to (2.46) One may then perform the field redefinition h s = λ s j s such that the new background field j s has the bare scaling dimension of a conserved current, (2.47) Considering the vicinity of the UV fixed point where 13 λ s → ∞, one gets , the Hubbard-Stratonovitch field σ s has been denoted h s (consistently with its bare scaling dimension) in order to stress that gauge symmetry is restored 12 Except possibly for very low spin s and dimension d which require a separate discussion (see e.g. [77] for detailed discussion of the double-trace deformation and its holographic interpretation in the s = 0 case). More precisely, for d = 4 the s = 0 term is marginal while for d = 2 the s = 0 term is relevant and the s = 1 is marginal. 13 Except possibly for low spin s and dimension d which might require a separate discussion which will be avoided here for the sake of simplicity.
in the limit λ s → ∞ due to the disappearance of the quadratic term. 14 Focusing on the logarithmically divergent piece, the left-hand-side in (2.48) can be interpreted as the generating functional of the correlators in CHS gravity. In particular, in the large-N (i.e. semiclassical) limit one has that the logarithmically divergent piece of The holographic dual to a Legendre transform on the boundary is a change of boundary conditions from Dirichlet to Neumann on the bulk fields. In fact, the solution space of the spin-s MHS fields with Neumann boundary condition is an so(2, d) module in one-to-one correspondence with the module of a spin-s shadow field (see e.g. [22] for a manifestly conformal and gauge invariant field-theoretical description). Consequently, the linearized spectra of off-shell CHS theory and Neumann MHS are in one-to-one correspondence. Therefore the result (2.42) can be rephrased purely in bulk terms as follows: d even: Neumann MHS = Higher-spin Algebra ⊕ Dirichlet MHS .
In other words, our character computation suggests that asymptotic charges account for all extra dynamical degrees of freedom in MHS theory when all boundary conditions are modified from Dirichlet to Neumann ones. Let us stress that both holographic duals to Dirichlet and Neumann MHS theories are (respectively, IR and UV) fixed points with unbroken conformal higher-spin symmetries (rigid symmetries in the former case, gauge symmetries in the latter) since all spins are on the same footing. As a side remark, one may observe that, the opposite sign in the last term on the righthand-side of (2.30) for d odd leads after summation over all spins to a relation between characters which one can rewrite as d odd: Dirichlet MHS ⊕ Neumann MHS = Higher-spin Algebra . (2.50) This relation suggests that the linearized MHS theory on AdS d+1 spacetime of even dimension without imposing any boundary condition (i.e. considering both normalizable and non-normalizable solutions) might also be somewhat "topological".

Extensions to Type-A ℓ and Type-B ℓ Theories
In the previous section, we have shown that the character of the type-A CHS theory coincides with the character of the adjoint module of the type-A higher-spin algebra. In this section, we provide more non-trivial evidences of this intriguing observation by generalizing the result to the type-A ℓ and type-B ℓ theories.

Field theory of type-A ℓ theory
Let us introduce another class of conformal gauge fields which are cousins of the spin-s FT field. They are described by totally-symmetric rank-s tensor h (t) s like the usual FT field, but they have weaker gauge symmetry [23]

1)
14 Therefore (2.48) would require a careful discussion of the the ghost contribution in the measure. This will not be performed here because we only intend to sketch the logic of the proof.
with ξ s−t a rank-(s − t) symmetric tensor, η the Minkowski metric and σ s−2 a rank-(s − 2) symmetric tensor. The integer t takes value inside the range 1 t s, parameterizes these class of fields and will be referred to as the depth (so that the usual FT field corresponds to t = 1). As in the usual FT field case, we can define a gauge-invariant field-strength, that is, a Weyl-like tensor as which also has conformal weight ∆ C (t) s,s−t+1 = 2 . In even d dimensions, the action for the depth-t and spin-s FT field is then given by After integrating by part, the action takes the form of where P s T t T is the t-ple transverse and traceless projector 15 which becomes local after multiplying by the factor s−t+1 . The condition δ ξ,σ h (t) s = 0 defines now the depth-t conformal Killing tensors.
An interacting theory of the depth-t FT fields can be obtained as an effective action, similarly to the t = 1 case. We replace the free scalar action by its analog of order 2ℓ in the derivatives and couple the system to the higher-spin sources h (t) s via a set of currents J (t) s which are traceless and t-ple divergenceless: One can show that for a given free scalar action with a fixed ℓ, we can find currents of all integers spin with t = 1, 3, . . . , 2ℓ − 1 [23,78]. These currents take the form where the ". . . " stands for additional terms ensuring (3.5), and it has the conformal dimension The tensors J (t) s with t > s do not satisfy any (partial-)conservation condition since (3.5) is not defined in such case. Still, these tensors can be used as part of the basis operators for the space of operators bilinear in the order-ℓ scalar singleton. When ℓ d 4 , all these operators are primary, and the space of operators with dimension s − 2k + d and spin s is spanned by the basis Starting from the 2ℓ-derivative scalar field action in the background of higher-spin fields of depths t = 1, 3, . . . , 2ℓ − 1, and integrating out the scalar field φ, we obtain an effective action. Again, the logarithmically divergent part of the effective action is a local functional of higher-spin fields, and for ℓ d 4 , it has the structure The functional W (ℓ) log can be regarded as an action of interacting higher-depth FT fields up to the introduction of a dimensionless coupling constant κ : log . This interacting theory contains not only the higher-depth FT fields but also other non-gauge conformal fields, referred to as special in [79]. We will refer to this class of fields as "special FT" for the sake of uniformity in the terminology. They correspond to the fields of spin-s and conformal dimension ∆ = 1 − s + t with t s + 1. Although they do not enjoy any gauge symmetry, we will keep referring to the parameter t defining those fields as their depth. In the quadratic part (3.10), the fields with 0 s 2 (ℓ − 1) and s+2 2 k ℓ correspond to this class. The free Lagrangians of these fields still has 2(s − t) + d − 2 derivatives but do not have any gauge symmetry. A trivial but important restriction to these fields is that the number of derivatives of their free Lagrangian cannot be negative. This gives a dimension dependent upper bound for t, namely t s + d−2 2 , and hence k d+2s 4 . The latter bound is irrelevant when it is not smaller than ℓ. The s = 0 case gives the lowest bound, and hence the condition that this be not smaller than ℓ imposes ℓ d 4 . From the CFT point of view, the bound implies that there is no operators with conformal dimensions lower than d 2 . This bound can actually be relaxed without encountering an inconsistency due to a subtle phenomenon discussed below. The origin of the quadratic Lagrangian in (3.10) is the local contact terms hidden in the two point functions of J (2k−1) s : One can note here that the contact terms are absent for k > d+2s 4 . This higher-depth CHS theory appears from the on-shell action of type-A ℓ MHS theory in AdS d+1 . Analogously to the usual FT/massless case, the depth-t FT fields in M d can be related to the depth-t partially-massless field in AdS d+1 : s ] + (regular or polynomially divergent terms).
(3.12) Here, Φ (t) s is the spin-s and depth-t partially-massless field and K (t) is the corresponding boundary-to-bulk propagator. Both the type-A ℓ theories, the CHS gravity in M d and partially-massless HS gravity in AdS d+1 , have the same global symmetries, the type-A ℓ HS symmetry algebra generated by higher-depth conformal Killing tensors.
When ℓ > d 4 , we face an interesting phenomenon referred to as extension on the CFT side in [78]. Let us briefly review this extension hereafter. For ℓ > d 4 , the operator spectrum contains pairs of currents J (2k−1) s and J (2k ′ −1) s with the same spin but with the respectively dual conformal dimensions, ∆ s,k + ∆ s,k ′ = d, in other words, For ℓ < d 2 , none of these currents are (partially-)conserved (see Fig.1), and the current operator with higher conformal dimensions becomes a descendent of the other: (3.14) Hence, the operators J (2k−1) s are both primary and descendent. Because of the degeneracy (3.14), we loose one basis operator from (3.8). As a consequence, we can find a new basis operatorJ (2k−1) s which is neither primary nor descendent. 16 Interestingly, the cross two point function of J (2k ′ −1) s andJ (2k−1) s does not vanish but gives  fields. It will be interesting to clarify how the "extension" phenomenon will affect the mechanism of double-trace deformation/changing boundary conditions. When ℓ d 2 , the solution space of ℓ φ ≈ 0 contains a conformally-invariant subspace. Hence, we are lead to consider either the finite-dimensional quotient part, or the subspace part. In fact, the quotient space corresponds to the space of harmonic polynomials with maximum order ℓ − d 2 . The zero mode of the ℓ = 1 and d = 2 case is the familiar example. The CFT based on the quotient part has been studied in [78]. We postpone the analysis of CHS theory in this case to a future work.

Character of type-A ℓ theory
The modules relevant to the type-A ℓ CHS theory are analogous to those of the ℓ = 1 theory: they are The character of the partially-conserved current module is simple: whereas the characters of the other modules are more involved and their explicit forms are given in Appendix A. What is important for our purpose is that these characters satisfy an analogous relation to (2.31), (3.19) Using the above properties together with the type-A ℓ Flato-Fronsdal theorem [23,83] we can handily compute the character of the entire type-A ℓ CHS theory. Here, the module V(s − t + d − 1; (s)) corresponds to a spin-s operator without any conservation condition. The field content of the type-A ℓ CHS theory consists of • the FT fields with depth t = 1, 3, . . . , 2ℓ − 1 and spin s = t, t + 1, . . ., and • the special FT fields with depth in the same range but spin s = 0, 1, . . . , t − 1.
We first sum the characters over the higher-depth FT field modules and obtain Again the second line vanishes due to the property and in the third line we find the character corresponding to the module of the special FT fields: Here, we used again the property (2.24) for even d. The modules D(1 + t − s; (s)) with t = s, s + 1, . . . arise from the "non-standard" BGG sequence of so(2, d), and they are associated with the special FT fields. This module exists in fact only if s t − d 2 + 2, which is the same condition that the special FT field action has positive derivatives. This condition is satisfied if ℓ < d 4 , and we obtain    Going off-shell.

Character of type-B ℓ theory
Finally, let us consider the type-B ℓ CHS theory based on the order-ℓ spinor singleton which will be denoted Di ℓ . It is related to the type-B ℓ MHS theory in an analogous way to the type-A ℓ case. Here, we focus on the case where the value of ℓ is smaller enough than the dimension d so that we do not encounter any subtle issue analogous to what we found in the type-A ℓ theory. The field content of the type-B ℓ CHS theory, can be read off from that of the the type-B ℓ MHS theory, which in turn is given by the corresponding Flato-Fronsdal theorem [83]: The universal factor 2 arises because we consider the parity-invariant Di module which possesses the pseudo higher-spin currents with γ d+1 insertion. Note that the character 2 χ D(s+d−t−1;(s,1 r−1 )) should be understood as χ D(s+d−t−1;(s,1 r−1 + )) + χ D(s+d−t−1;(s,1 r−1 − )) . This theory contains infinitely many higher-spin gauge fields corresponding to the modules D(s + d − t − 1; (s, 1 m )) with s t + 1 − δ m,0 , whose Killing tensors form the type-B ℓ higher-spin algebra. Note that for s < t + 1 − δ m,0 , the generalized Verma module is irreducible: − 1; (s, 1 m )) . The vector space of the type-B ℓ higher-spin algebra can be decomposed into so(2, d)-modules as [63] Type (3.31) The character of the conformal Killing tensors appearing above is related to the character of the corresponding FT field module in M d through where m 1 and s t + 1. If the latter condition is not satisfied, then we get the special FT field module: We sum the characters (3.32) and (3.33) together with the m = 0 counterparts (3.19) and (3.23) over the field content of type-B ℓ CHS theory (3.29), and use the Flato-Fronsdal theorem (3.30) to simplify the series. In the end, we find the relations (1.1) and (2.42) hold also for the type-B ℓ theory.

Discussion
In this paper, we calculated the so(2, d) character of CHS gravity spectrum and showed that it coincides with that of its global symmetry algebra, namely (conformal) higher-spin algebra. The evaluation of the full character requires the summation of the characters of each field appearing in the spectrum of CHS gravity. The character of each conformal gauge field satisfies the relation (1.2), which provides a simple link to its associated conformal Killing tensor, the collection of which span the CHS symmetry algebra. Our key observation is that, in the character of each gauge field, the contributions besides the Killing tensor cancel out after the summation. Below, let us make two remarks: first about the key relation (1.2), then about the summation. The relation (1.2) -whose concrete versions for partially-massless fields of symmetric and hook-symmetry types are given respectively in (3.19) and in (3.32) -is in fact very general. In Appendix A, we established the relation for partially-massless fields of any depth and any symmetry: see (A.44). Note that this relation is valid also for non-gauge fields -e.g. for totally-symmetric and hook-symmetry type fields we have (3.23) and (3.33) -where only the Killing tensor contribution is absent: see (A.40) for the most general case. Another way to state this property is that the character of any d-dimensional conformal field, irrespective of whether it has gauge symmetry or not, is given by the difference of the characters of the associated field in AdS d+1 with different boundary conditions. The character for the field with Neumann condition is the same as the character with Dirichlet condition but q replaced by q −1 , plus -if the field has gauge symmetries -the character of the Killing tensor. Therefore, the character of a theory composed of multiple conformal fields is always given by the character of its global symmetries and the rest which is the difference of the character of the AdS d+1 counterpart theory under the flip q → q −1 . Hence, when the character of the AdS d+1 theory is symmetric under q → q −1 , only the character of the global symmetries survives as is the case for type-A ℓ and type-B ℓ . For this reason, our observation extends neither to the type-C case nor to the minimal theories, but holds for the type-AZ theory [84].
Let us turn to the issue of the summation over spins or field contents. Since we have just equated, as series, the character of CHS gravity and that of its symmetries, we did not elaborate on whether this series is convergent or not. In fact, there are several issues. First, the character of the AdS d+1 theory is given by a series which is convergent inside the disk |q| < 1. It shows the symmetry q → q −1 only after an analytic continuation, and hence the cancellation of these terms already involves a regularization. Next, the character of the higher-spin symmetries is given by a series which does not have any region of convergence. Nevertheless, we can still evaluate this series dividing it into the pieces having different regions of convergence, and this procedure makes sense only in terms of distributions (see [64] for more discussions). After these regularizations, we obtain the character of CHS gravity as a function of q and x . For instance, for the type-A CHS theory in d = 4 we find (4.1) Since the one-loop partition function is the character evaluated at x = 1, we can attempt to get the CHS partition function from this character. Since the limit x 1 , x 2 → 1 of the last two terms of (4.1) is singular, we would need to subtract the finite part in the limit. However, this procedure is somewhat ambiguous as it depends on how we take this limit. Moreover, the simplest trials (e.g. sending x 1 and x 2 consecutively or simultaneously to 1) do not easily reproduce the result found in [54]: The above was obtained by summing the partition functions of each FT field (i.e. their individual characters evaluated in x = 1) with a damping parameter ǫ, i.e.
Here C is a contour encircling the origin of the complex ǫ-plane, so the contour integral picks up the finite part of the integrand. The shift d−3 2 in the e −ǫ(s+ d−3 2 ) regularization is introduced to ensure the vanishing of the a-anomaly coefficient and the Casimir energy of CHS gravity [36,65]. As already detailed in [54], a closer investigation reveals that the subtleties lie in the first term of the on-shell FT field character, (4.4) because the second and third terms in the above expression lead to convergent series as we sum over the spins. The first term is the character of the generalized Verma module with lowest weight [2; (s, s) 0 ], so the spin-dependent part is simply the so(4) character, Therefore, we see that the issue is in fact how to sum the above character evaluated at x 1 = x 2 = 1 over all spins. For a concrete understanding, let us consider The above function is regular in either limit x → 1 or ǫ → 0 but not the simultaneous limit. And depending on the evaluation order, we indeed obtain different results. Setting x to 1 and then extracting the finite part in the ǫ → 0 expansion leads to the partition function (4.2), whereas setting first ǫ to 0 and then extracting the finite part in the x → 1 expansion leads to It is interesting to note that the above partition function also gives vanishing Casimir energy. The cancellation of the a-anomaly of CHS gravity can be also shown using the characters (see Subsection 2.4). Let us conclude by a few comments on CHS theories around an AdS background. The kinetic operator of spin-s Fradkin-Tseytlin fields around AdS d factorize into a product of kinetic operators of spin-s partially-massless fields with all possible depths (together with a finite number of spin-s massive fields in dimensions d 4) [29,30,34]. Notice that this property was recently generalized to the case of higher-depth Fradkin-Tseytlin fields in [31]. From a group-theoretical perspective, the factorization of the kinetic operator of Fradkin-Tseytlin fields is related to the branching rule of the corresponding module: as detailed in Appendix C, the on-shell Fradkin-Tseytlin module branches into the so(2, d−1) modules which correspond to the kernels of each factor of the Fradkin-Tseytlin kinetic operator. 17 Consequently, the CHS gravity around an AdS background can be viewed as a local interacting theory of massless, partially-massless and massive (with "the mass squared" smaller than that of massless fields) fields of arbitrary spins. Such a field content as an AdS d theory can be obtained by simply branching the representation carried by the CHS gravity spectrum from so(2, d) onto so(2, d − 1). For instance, in d = 4 dimensions the type-A CHS theory spectrum branches onto the direct sum of partially-massless fields in AdS 4 with all integer spins and depths. 18 Taking a definite boundary condition 19 we find that the so(2, 3) character of this theory reads where we have written the character in terms of the temperature β and the variable α defined through q = e −β and x = e iα . Using the method introduced in [66,67], we can compute the one-loop free energy of this theory in Euclidean AdS 4 using the above character. To do so, one needs to evaluate the character (4.8) and its derivatives (with respect to the variable α) in α → 0, which is singular here (contrary to MHS theories and their partially-massless cousins). As a consequence, one has to introduce a new regularization scheme to cure this additional source of divergence. We can easily think of two possibilities: • To extract only the finite part of (4.8) in its α → 0 expansion which leads to the one-loop free energy (4.9) • To introduce a damping factor in the summation of the character, so that the resulting character has a finite limit in α → 0. One can then extract the finite part in the ǫ → 0 expansion and compute the one-loop free energy using the resulting expression. Using a shift γ = 1 2 as in the case of MHS or CHS theory in four dimensions, we end up with (4.11) None of the results (4.10) and (4.11) is particularly convincing, and hence we would need clearer guidelines for the regularization from other inspections. This naturally calls for a better understanding of CHS gravity from the point of view of an "all-depth" partiallymassless higher-spin theory in AdS d . In fact the latter perspective is interesting to explore 17 A noteworthy subtlety here is that here appear both of the modules corresponding to the solutions with Dirichlet and Neumann boundary conditions. 18 When d 6, the branching rule will also contain an infinite number of massive fields with arbitrarily high integer spin. 19 When d = 4, the partially-massless fields with Dirichlet and Neumann boundary conditions correspond to isomorphic modules.
in its own right and there are several amusing questions. The first question is whether CHS gravity can afford a truncation like the one from Weyl gravity to Einstein gravity [28]. The truncation to MHS fields would not be consistent since there is no subalgebra of the CHS algebra compatible with the truncated spectrum, but the truncation to all odd-depth fields may lead to a consistent theory [85]. The second question is what might be the CFT d−1 dual of CHS gravity viewed as an exotic partially-massless higher-spin theory in AdS d . It cannot be one of the usual free CFTs as they do not exhibit such operator spectra, but their exotic cousins might be decent candidates: in [85] the non-local free scalar CFT with the kinetic operator √ was proposed as the dual of the odd-depth truncation of CHS gravity in AdS. The third question is whether CHS gravity can be defined as a local theory in certain odd dimensions. It is well known that three-dimensional CHS gravity can be realized as a Chern-Simons theory. Therefore, it is natural to inquire whether the higher-dimensional Chern-Simons actions may prove useful for the local realization of some parity-breaking CHS gravity in odd dimensions. Lastly, assuming this works, one might then speculate whether one can iterate the construction [55]: MHS gravity in AdS d+1 leads to CHS gravity in AdS d , which in turn could lead to another class of CHS gravity in AdS d−1 , etc.

A Zoo of modules
In this appendix, we review the description of the various fields of interest for this paper in terms of the corresponding so(2, d)-modules, as well as the associated characters. To do so systematically, we go through the BGG resolution of these modules, following the discussions in [47,54,86] (see also [58,87] for more details on the relevant characters from a different perspective).
• AdS d+1 gauge field module / CFT d current: The module defined by the lowest weight corresponds to a partially-massless mixed-symmetry field with spin Y = (s 1 , . . . , s r ) and minimal energy ∆ The presence of this submodule signals the fact that these partially massless fields are subject to gauge symmetries, namely the submodule of lowest weight [∆ (t) Y p,t := (s 1 , . . . , s p−1 , s p − t, s p+1 , . . . , s r ) .
The resulting quotient, can be realized as the space of traceless and divergenceless tensor with symmetry Y and solution to the wave equation where ξ Yp,t is also a traceless and divergenceless tensor with symmetry Y p,t and subject to a wave equation similar that of ϕ Y (see [88][89][90][91][92][93] for more details). When p > 1, the gauge symmetry (A.6) is reducible, i.e. it is trivial for some particular class of gauge parameters. This implies that the module D ∆ (t) p +t ; Y p,t of gauge parameters is itself a quotient, obtained by modding out of V ∆ (t) p + t ; Y p,t the irreducible submodule describing the parameters leading to trivial gauge transformations. For a generic partially-massless as we are considering here, there are p classes of "gauge parameters", the genuine one with diagram Y p,t and (p − 1) reducibilities, which are obtained from the BGG resolution. Denoting Y (k) p,t the Young diagram describing the kth of these reducibility parameters, they can be expressed in terms of Y p,t by with again the convention that ν 0,p = 0, the minimal energy of these reducibility parameters reads Schematically, each reducibility parameter is obtained from the previous one by removing more and more boxes. Starting from the gauge parameter, where t boxes are removed from the pth row, one then obtain the first reducibility parameter by removing boxes from the row above (the (p − 1)th one) until this row has one less box than the one below (the pth one) in the original Young diagram (i.e. s p − 1). Applying the same procedure to the obtained reducibility parameter, one can construct the next one and so on.
From the d-dimensional point of view, these modules describe (partially-)conserved currents of spin Y, i.e. operators of conformal weight ∆ (t) p which are traceless tensors with the symmetry properties of the Young diagram Y. The conservation law obeyed by these operators is of the form where P Yp,t T is a traceless projector onto Y p,t . In this context the submodules involved in the definition of this irreducible module are interpreted slightly differently than in the AdS d+1 case. The first submodule is spanned by the conservation law (A.11), i.e. a suitable projection of a (or several) divergence(s), and all its descendants. The remaining submodules correspond to descendants of the conservation law which are identically 20 vanishing (i.e. for symmetry reasons).
We can now write down (irrespectively of the parity of d) the character of the module gauge field module D ∆ (t) p ; Y as follows: • Curvature of the gauge field in AdS d+1 : The module whose lowest weight reads object as the symmetry property of the Young diagram Y c which is obtained from Y by adding s p − s p+1 − t + 1 to the (p + 1)th row Y c = (s 1 , . . . , s p , s p − t + 1 ↑ (p+1)th , s p+2 , . . . , s r ) . (A.14) Schematically, the curvature is given by where P Yc T is the projector onto the symmetry of the Young diagram Y c (without any tracelessness condition). Accordingly, the minimal energy of the module of the curvature tensor is given by 16) i.e. the minimal energy of the gauge field ϕ Y added by the number of derivative acting on it to make up the curvature. The character of this module reads (irrespectively of the parity of d) • Conformal Killing tensor: The finite dimensional module whose BGG resolution contains the above gauge field module corresponds to that of its conformal Killing tensor. In AdS d+1 , this corresponds to the space of solution of where P Y projects onto the symmetry of the Young diagram Y without any tracelessness condition, whereas in M d this module corresponds to the space of solution of the conformal Killing equation In terms of so(2, d)-weight, the conformal Killing tensor module has lowest weight  p − ∆ c = s p − s p+1 − t + 1 derivatives on φ Y , projected so that the resulting tensor has the symmetry of Y c . Schematically, we have where P Yc T is the traceless projector onto the symmetry of the Young diagram Y c . The character of this module reads, for d = 2r: and for d = 2r + 1: Notice that in both cases, characters of modules labelled by the so(d)-weights Y (m) c correspond to the generalized Bianchi identities verified by the Weyl tensor.
• Off-shell Fradkin-Tseytlin field in M d : The off-shell Fradkin-Tseytlin, or shadow, field φ Y associated to the (partially) massless field ϕ Y is a field of conformal dimension and spin Y, subject to the gauge transformation where P Y project onto the symmetry of Y whereasY denotes any Young diagram obtained by taking a trace of Y, so that σY should be understood as a collection of Weyl gauge parameters with the symmetry of all possible traces of Y. As already mentioned previously, this field does not correspond to a generalized Verma module and consequently is not found in the BGG resolution in even dimension. However, as proposed in [54], the corresponding character can still be computed by translating the field-theoretical definition of this conformal gauge field. In even d dimensions, we therefore define where: -The first term, which is the character of the generalized module with lowest weight (d − ∆  38) in accordance with the fact that the module of the shadow field is isomorphic to that of the off-shell Weyl tensor.
Series of modules with non-integral weight. Another class of irreducible modules, with non-integral weight, can exists and are defined by the quotient The character of the module (A.39) is given by The scalar and the spinor order-ℓ singletons (respectively, Rac ℓ and Di ℓ ) are part of this class of module [23], as their conformal weight read with s = 0 or s = 1 2 (i.e. the spin of these fields). It is clear from the above conditions that in odd dimension d, these two families of singletons exhaust the fields described by this class of module. However, for d even one can consider more general conformal fields (in particular they can be of spin greater or equal to one). In general, the above class can be realized as conformal fields φ Y of spin Y obeying to a wave equation of the form: Notice that even if Y = 0, these fields do not enjoy any gauge symmetry (contrarily to, for instance, the usual CHS fields sourcing the conserved current in (2.7)).
On top of that, one can derive the following relation between (A.12), (A.29) and (A.34) which, upon using the previously derived (A.43), also leads to Recalling that for odd d the character of the Weyl module is identical to that of the shadow field, we can write the following relation valid in any dimension: The above relation allows to express the character of the shadow field in terms of the character of the associated partially massless field, plus or minus the character of their conformal Killing tensor (depending on the symmetry of these fields).
Action principles in the metric-like formulation. Let us conclude this appendix by commenting on the action principle for a free conformal field of spin Y and conformal weight d − ∆ (t) p . Schematically, it takes the form (A.48) Notice that the number of derivatives involved in this action can be constrained by requiring the latter to be quadratic in the shadow field with spin Y, and that the integrand has conformal weight d. A careful analysis of such action principles can be found in [13], where in particular the form of the action (A.48) is derived by requiring it to be conformally invariant (see also [79] for the case of totally symmetric fields).

B Two-dimensional case
The d = 2 conformal algebra is a direct sum of two d = 1 conformal algebras: so(2, 2) = so(2, 1) ⊕ so(2, 1) corresponding to left and right movers if we write the flat metric in light-cone coordinates x ± on M 2 as ds 2 = 2 dx + dx − . Accordingly, the spin label can have positive/negative real values corresponding to left/right chirality. The so(2, 2) Verma modules will be denoted as V(∆; s) and are related to the so(2, 1) Verma module V w with lowest weight w as Introducing the complex variables, for the so(2, 2) weights, the character of V(∆; s) is given by The chirality-invariant modules will be denoted with 0 as subscript, e.g. V(∆; s) 0 := V(∆; +s) ⊕ V(∆; −s) . for the two chiralities, where 1 stands for the trivial representation of so(2, 1) and we made use of (B.1) and V 0 /V 1 ∼ = 1 . Accordingly, their respective characters are The above can be straightforwardly generalized to the partially-conserved currents: (see e.g. Section 6 of [94]). ± are densities of weight w in one dimension described as the so(2, 1) Verma module V w . The spins s < 2 require a more careful discussion (see e.g. Section 4.2 of [64]) so we will just mention that Di = D( 1 2 ; 1 2 ) 0 describes the conformal spinor while D(1; ±1) describe a left/right chiral boson J ± = ϕ (1) ± (x ± ) which both solves the Klein-Gordon equation ∂ + ∂ − J ± = 0 (so it is a submodule of Rac) and the conservation law ∂ − J + + ∂ + J − = 0 (so it is a submodule of the complete module describing the spin-1 conserved current). In fact, the shadow fields are pure gauge in d = 2 and the negative sign corresponds to the residual gauge symmetries. 21 Consider a shadow field h s of spin s 2, one can first reach the traceless gauge where all components of the type h +−··· vanish. The residual gauge symmetry reads δ ξ h ±···± = ∂ ± ξ ±···± and allows to fix the reach the trivial gauge h s = 0. Even after this complete gauge-fixing, the parameters of the residual gauge symmetry are functions of one variable: . This is nothing but the higher-spin version of the usual infinite-dimensional enhancement of conformal symmetries in d = 2.
Using (2.24) for d = 1, one can obtain the relation χ so(2,1) The contragredient of the so(2, 1) lowest-weight Verma module V ∆ is the so(2, 1) highestweight Verma module with highest weight −∆, which we will denote V ∆ . Its character is χ . Hence, it may be tempting to reinterpret the character formulae for the spin-s shadow fields in terms of the one for the contragredient so(2, 1) Verma module with highest weight s − 1. In this sense, one can write the formal equality: S(s; +s) = 1 ⊗ V −(s−1) and S(s; −s) = V −(s−1) ⊗ 1.
The list of the relevant parity-invariant so(2, 2)-modules and their field-theoretical interpretations are summarized in Table 2.

) modules and their field-theoretical interpretations
Conformal higher-spin gravity. Conformal higher-spin gravity is highly degenerate in two dimensions since its action identically vanishes for s 2 in a topologically trivial background while the sector of low spin (s = 0, 1) is non-local. Nevertheless, its partition function can be defined (see [54,95]). As far as characters are concerned, one can write d = 2 versions of (2.41). They rely on the d = 2 Flato-Fronsdal theorems (cf. section 4.2 in [64]). For instance, in the simpler type-B case, One may define formally the character of the parity-symmetric spin-s on-shell FT module as the analytic continuation of the one for the quotient in the right-hand-side of (2.18) for d = 2, i.e. the difference χ so(2,2) D(s,s) 0 . By summing over all spins, one can say that (1.1) also holds in d = 2.

C Branching rule for the Fradkin-Tseytlin module
In this appendix, we derive the branching rule for the so(2, d)-module of the on-shell FT field in even dimension d = 2r, i.e. we decompose this irreducible so(2, d)-module into a direct sum of so(2, d − 1)-modules. The latter can naturally be interpreted as fields in AdS d , thereby allowing us to recover the factorization property of the kinetic operator of CHS fields in M d into a product of kinetic operators of partially-massless and massive fields in AdS d obtained in [29,30,34].

C.1. Branching rule for generalized Verma modules
In order to derive the branching rules for on-shell Fradkin-Tseytlin modules, we first need the following lemma [96,97]: The idea of the proof (for more technical details, see e.g. [96,97]) is simple enough, when expressed in terms of the Poincaré-Birkhoff-Witt basis, so that we recall it here, before giving another proof in terms of characters.
Poincaré-Birkhoff-Witt basis. The generalized Verma module V(∆; Y) defined by (C.1) is constructed as follows: one starts from the module V [∆ ; Y] which is an irreducible representation of the homothety subalgebra p d := ( so(2) ⊕ so(d) ) R d in which the action of R d is trivial. Concretely, the conformal algebra admits a three-grading decomposition so that (as a vector space), where K a are the special conformal transformation generators, P a the translation generators, M ab the so(d) generators and D the dilation generator (while a, b = 1, . . . , d).
The vector space V [∆;Y] carries a representation of p d in the sense that so(2) acts diagonally with ∆ as eigenvalue, so(d) acts through the representation Y = (s 1 , . . . , s r ) while R d ∼ = g −1 ∼ = span(K a ) acts trivially. In other words, it defines a primary field, i.e. a field with fixed conformal weight ∆ and spin Y annihilated by special conformal transformations at the origin. The generalized Verma module V(∆; Y) is then constructed by acting with the whole universal enveloping algebra U so(2, d) on the finite-dimensional module V [∆;Y] , modulo the action of p d . In practice, the action of (arbitrary powers of) the generators of Y] defines new vectors (or states) in the generalized Verma module, so that The action of the conformal algebra on V(∆ ; Y) is then defined by the action of the subalgebra p d on V [∆;Y] together with the Lie bracket between generators of this subalgebra and of g +1 . In other words, the representation of so(2, d) is induced from representation of the subalgebra p d . In more concrete terms, we can write a basis of V(∆; Y) as | ∆ + n ; Y a 1 ...an = P a 1 . . . P an | ∆ ; Y , (C.7) where | ∆ ; Y stands for a generic basis element of V [∆;Y] . In order to obtain the branching rule of V(∆; Y), the first task is to branch the representation V The next thing to do is to decompose the action of U (g +1 ) ∼ = U (R d ) on the direct sum of the representation of p d−1 into a direct sum of so(2, d − 1) generalized Verma modules. To do so, let us single out one generator of g +1 , say P d , so that the remaining generators P i (i = 1, · · · , d − 1) span R d−1 . The basis of V(∆; Y) now reads P i 1 . . . P ip (P d ) n | ∆; σ , n, p ∈ N , σ ∈ B(Y) , (C.10) where the indices i 1 , . . . , i p run from 1 to d − 1, while | ∆; σ stands for a generic basis element of the finite-dimensional p d−1 -module V [∆;Y] labeled by the so(2)-eigenvalue ∆ and so(d − 1)-weight σ, and B(Y) designates the set of so(d − 1)-weights σ in the above branching rule of the so(d) irrep Y. Due to the fact that P d has weight +1 under the adjoint action of the dilation generator (as any generator of g +1 ), and commutes with any element of so(d − 1), we have (P d ) n | ∆; σ ∼ = | ∆ + n; σ , (C.11) i.e. it defines an irrep of p d−1 with so(2)-weight ∆ + n and so(d − 1)-weight σ . Then the branching rules (C.3)-(C.2) of the conformal algebra result from the branching rules (C.8)-(C.9) of the rotation subalgebra.
• For d = 2r, the rule (C.2) is less straightforward to recover at the level of character due to the fact that the rank so(2, d) is r + 1 whereas the rank of so(2, d − 1) is r, and hence the characters of modules of the latter depend on one less variables than the characters of the former. This can be already observed for the branching of so(d) irreps (C.15) which at the level of characters reads , (C.17) where δ(X 1 , . . . , X r ) is the r × r Vandermonde determinant. In order to simplify (C.15), one can set one of the variables x of the so(d) character to 1, say x r , so that the characters on both sides of this equations depend on the same number of variables. Due to the fact that The characters appearing on the right-hand-side of the above equation correspond to the following modules: • For t = 1, . . . , s, the characters χ so(2,d−1) D(s+d−t−2;(s)) (q, x r ) are those of the modules describing partially-conserved currents of spin-s and depth-t in d − 1 dimensions (given in (3.18)), while χ so(2,d−1) S(1+t−s;(s)) (q, x r ) are the characters of the corresponding spin-s and depth-t shadow fields in M d−1 . Notice that the former/latter modules can also be interpreted as AdS d fields, namely as spin-s and depth-t partially-massless fields with Dirichlet/Neumann boundary behavior.
Having in mind that the so(2, d − 1)-modules S(s + d − t − 2; (s)) and D 2; (s, s − t + 1) coincide for t = 1, . . . , s and correspond to the off-shell FT module either in terms of the shadow field h s or in terms of its Weyl tensor C s,s , we can write the branching rule: This decomposition is in accordance with the results of [29,30,34] (conjectured in [36]) where the decomposition of the CHS wave operator in R d into a product of wave operators in AdS d for the partially massless fields of depths t = 1, · · · , s . Firstly, the left-hand-side describes the on-shell FT field on d-dimensional conformally flat space M d as a module of the conformal algebra so(2, d). Secondly, the right-hand-side correspond to its description in terms of partially massless fields on AdS d as modules of the isometry algebra so(2, d−1). Thirdly, each partially massless field gives rise to two modules: one for each possible boundary choice. Following similar observations in [55], let us rephrase their heuristic argument as follows: The conformal boundary of Euclidean 22 AdS d+1 is the conformally-flat sphere S d . In turn, Euclidean AdS d is conformal to one hemisphere of S d with two possible choices of boundary conditions at the equator S d−1 . Therefore, defining a conformal field on S d in terms of the one on Euclidean AdS d we need to sum over the two boundary condition choices. Notice that the case d = 4 is special, as no massive nor shadow fields appears in the branching rule, so that (C. 27) In fact, the modules D 2; (s, s − t + 1) are absent since finite-dimensional irreducible so(3)modules labelled by two-row Young diagrams vanish (and hence these modules do not exist for so (2, 3)).
Higher-depth Fradkin-Tseytlin modules. More generally, for a depth-t FT field, the branching rule for d = 4 reads i.e. partially massless fields of different spins (as well as different depth) appear. Notice that this is in accordance with the factorization of the partition function of maximal depth FT fields derived in [56]. In higher dimensions (d = 2r 6) the branching rule also involves more modules (as in the t = 1 case), namely Notice that this branching rule contains 2 t (s − t + d−2 2 ) modules whereas the kinetic operator for the FT field contains s − t + d−2 2 factors (see e.g. [31]). The kernel of each factor operator, labelled by k ranging from 0 to s − t + d− 4 2 , corresponds to the module s σ=s−t+1 S 1 + t + k − s; (σ) ⊕ D s + d − t − k − 2; (σ) . For instance, the d = 4 maximaldepth (s = t) FT fields have two-derivative kinetic operators but their spectrum is made of maximal-depth PM fields of spin 1 to s (see Section of 3.4 of [98] for the concrete example of s = t = 2 case).