On the conformal higher spin unfolded equation for a three-dimensional self-interacting scalar field

We propose field equations for the conformal higher spin system in three dimensions coupled to a conformal scalar field with a sixth order potential. Both the higher spin equation and the unfolded equation for the scalar field have source terms and are based on a conformal higher spin algebra which we treat as an expansion in multi-commutators. Explicit expressions for the source terms are suggested and subjected to some simple tests. We also discuss a cascading relation between the Chern-Simons action for the higher spin gauge theory and an action containing a term for each spin that generalizes the spin 2 Chern-Simons action in terms of the spin connection expressed in terms of the frame field. This cascading property is demonstrated in the free theory for spin 3 but should work also in the complete higher spin theory.


Introduction
The purpose of this paper is to propose a set of field equations in three dimensions that describe a fully interacting conformal system consisting of a scalar field and the higher spin theory generated by the SO(3, 2) higher spin algebra. We will follow the work [1] where this approach to the problem was discussed in some detail and some results were found indicating that this may be worth pursuing further.
After giving the proposed equations in section 2 we will first explain the notation and content of them and then present some of the arguments leading to their particular form together with some explicit checks that will give some support for this proposal. The higher spin part of the theory has its origin in a gauged SO (3,2) Chern-Simons theory which can be reformulated as a generalization to all higher spins of the standard spin 2 Chern-Simons theory for the spin connection. This will be elaborated upon in section 3 where a cascading trick is used to relate the two different Chern-Simons formulations of the higher spin theory. Some additional comments are collected in the Conclusions.

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Note, however, that these equations are taken from the topological gauging of three dimensional CF T s with eight supersymmetries [2,3] where all coupling constants are determined in terms of the gravitational one g. This needs not to be the case in non-supersymmetric theories like the ones we deal with here. Once the higher spin equations are presented we can discuss their consequences for the field equations for spin 3 and above. Only a few such comments will be given here while a more extensive survey will be left for a future publication. We may note already at this point that the spin 2 equation above will be augmented by new terms with more than two derivatives of the scalar φ(x) provided higher spin frame fields also appear. This is true also for the equations of spin 3 etc and follows directly from the fact that φ has conformal dimension L − 1 2 and that the number of derivatives in the spin s equation is 2s − 1 which implies that the spin s frame field itself is of dimension L s−2 . Also the Klein-Gordon equation will contain terms with higher spin fields and more than two derivatives on the scalar.
We will find it convenient to write the Cotton equation (1.2) in irreps of SO (1,2). The point is that the trace is exactly the Klein-Gordon equation which means that the rest of the Cotton equation is in the irrep 5 and reads where we recall that the Cotton tensor is already in this irrep. The purpose of this paper is to suggest two higher spin field equations containing component equations for all spins ≥ 2 coupled to a scalar field φ with φ 6 potential and which in particular reproduce both the above Klein-Gordon and spin 2 Cotton equations. Already in [1] where this approach was discussed, but without source terms in either equation, it was shown that the correct curvature scalar term does arise in the Klein-Gordon equation and in addition also a spin 3 contribution 1 φ − 1 8 Rφ +f φ = 0, (1.4) where the spin 3 term contains the tracef := e µ afµ a (1,3). The fieldf µ a (1,3), which is an expression containing three derivatives on the spin 3 frame field e µ ab is discussed briefly later in this paper. The reader is advised to consult [1] for definitions and more details on the spin 3 sector of the higher spin system. The problematic issue of constructing source terms was mentioned in this context at the end of that paper, and a suggestion how it can be solved is presented in the next section. The higher spin algebras together with the linearized versions of the zero field strength and unfolded equations have been discussed in many papers in the past, see, e.g., [4][5][6][7][8][9][10] and referencies therein. The conformal higher spin sector of the theory that is the subject of this paper is also analyzed in a recent paper by Vasiliev [11] where its relation to higher spin theory in AdS 4 is used to draw conclusions about Lagrangians etc. The scalar sectors, on the other hand, are not the same. In fact, the scalar considered in this paper is the one discussed in [9]. The linearized spin 3 frame field system used in section 4 below is JHEP08(2016)142 discussed in the "metric" formulation in, e.g., [12]. Furthermore, the explicit analysis of the conformal higher spin system performed in this paper is closely related to the more formal approach of σ − cohomology developed by Shaynkman and Vasiliev, see [13][14][15]. 2

The conformal interacting higher spin equations
The two basic field equations for the SO(3, 2) conformal higher spin (HS) theory coupled to a scalar field with fifth order self-interactions that we propose and study here are the unfolded equation where D = d + A, and the following field strength equation valued in the SO(3, 2) higher spin algebra In the first subsection below we explain the notation used in these equations following [1]. The remaining two subsections are then devoted to a more detailed study of the sourced equations.

2.1
The higher spin setup: review of the F = 0 and DΦ|0 q = 0 equations The HS field strength F = dA + A ∧ A is obtained from the HS gauge field A which has the following expansion To understand the structure of this gauge field we give the parts of the higher spin conformal system that will explicitly play a role below, namely the spin 2 part where the generators of translation, Lorentz, dilatation and special conformal transformations are, respectively, P a , M a , D and K a with their associated gauge fields e a , ω a , b and f a , and the spin 3 part A 2 = e ab P ab +ẽ abP ab +ẽ aP a +ω abM ab +ω bM a +bD +f aK a +f abK ab + f ab K ab . (2.5) The gauge fields (lower case quantities) and generators (upper case) of the HS algebra appearing in these expressions are all in irreps, i.e., the a 1 . . . a n are totally symmetric and traceless sets of three-dimensional vector indices. The fields e a 1 ...an are the spin s = n + 1 frame fields and we will call f a 1 ...an the Schouten tensor 3 since Df a 1 ...an + . . . = 0 turns out to be the spin s = n + 1 Cotton equation which is of order 2s − 1 in derivatives. 2 An entirely different method to construct a conformal higher spin theory in three dimensions was suggested by Segal in the work [16] which, however, does not give rise to field equations of the Cotton type as is the case of the work presented here. 3 A perhaps more appropriate definition of the Schouten tensor is used, e.g., in [12], which corresponds to the spin 3 fieldf ab in (2.5).

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We emphasize here that all the fields depend only on the three dimensional space-time coordinates x µ and there are thus no dependence on any other coordinates or auxiliary variables like the ones 4 often appearing in Vasiliev's constructions of interacting higher spin theories in AdS.
The HS algebra can be defined as follows. Consider the so(2, 1) ≈ sp(2, R) spinor variables q α , p α (with α, β, .. = 1, 2) which are hermitian operators satisfying [q α , p β ] = iδ α β . The spin s = n + 1 HS generators are then given by all Weyl ordered polynomials in q α , p α of degree 2n. For example, for s = 2 we have and for s = 3 By computing the algebra of these generators keeping only single commutator terms we obtain the classical higher spin algebra based on the Poisson bracket used in this context in the original work on conformal higher spins in three dimensions [6]. Instead, by quantizing the variables q α , p α and Weyl order them as above they generate the for us relevant higher algebra of SO (3,2). Note that for all generators G(2n) (which are of order 2n in q α , p α ) with n vector indices the ordering of the q and p operators do not matter and they are thus automatically ordered as required. We give the operators q α dimension L 1 2 and p α dimension L − 1 2 which means that both A and F will be dimensionless. This will be useful later when we discuss how to construct S and T on the r.h.s. of (2.1) and (2.2). With these rules all multiplications can be viewed as star products which, however, has to be remembered since it is not explicitly shown by our notation.
We now turn to the l.h.s. of (2.1). The derivative operator appearing there is just D = d + A where A is as defined above. 5 However, the scalar field Φ(x) is special and differs in its definition from A. Φ is expanded only in terms of the most special conformal generators K a 1 ...an which is the last one of the generators in each spin s = n + 1 field A n above and contains only the variables p α , in fact, exactly 2n of them. We define the HS scalar field as follows where the first term defines the usual scalar field φ(x) that will appear conformally coupled to spin 2 and all higher spin frame fields coming from A and with its own fifth order selfinteraction in the Klein-Gordon equation.

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The vacuum used in (2.1) is defined to be annihilated by the q α operators making it translationally invariant in the sense that P a |0 q = 0. Although Φ itself does not contain any q α operators, the fact that A does will lead to the appearance of interaction terms already for spin 2. In particular, a correctly normalized Rφ interaction term appears directly after starting the unfolding procedure as observed in [1]. The scalar field is conformal and thus of dimension L − 1 2 so the l.h.s. of (2.1) is a one-form of dimension L − 1 2 which must be true also for S on the r.h.s. of that equation. We will propose an expression for S below after explaining the structure of the second equation F = T .
The role of the vacuum in the unfolded equation (2.1) is clear and a well-known property of this kind of scalar field, see, e.g., [9]. However, one of the crucial points in this discussion is to understand the relation of the two field equations (2.1) and (2.2) where the former one involves the vacuum while the latter one does not and hence has components for every generator of the higher spin algebra. To make the following argument a bit more explicit we give the spin 2 and spin 3 equations coming from the generator decomposition of F = 0. Note, however, that the following spin 2 and 3 equations have been truncated to the single commutator terms for simplicity.
For spin 2 the equations are (in the gauge b µ = 0) [5] e a ∧ f a = 0, (2.14) Df a = 0, (2.15) where T a = De a = de a + ω a bc ω b ∧ e c and R a = dω a + 1 2 ω a bc ω b ∧ ω c . The second of these is useful for us since solving it for f µ a gives where S µν is the Schouten tensor. The third equation in the above F = 0 spin 2 system then just says that the Schouten tensor is symmetric and the last equation that it satisfies the Cotton equation. For spin 3 we give only the cascading equations (see the next section) used to express the spin 3 Schouten tensor f µ ab in terms of the frame field e µ ab (the full system including the constraint equations is discussed in [1,19]) From the explicit structure of these equations (and the work in [1,6]) it should be clear that they can all be solved algebraically except for the very last one which is the Cotton equation and that this works for all spins. The result of this procedure thus expresses the JHEP08(2016)142 spin s = n + 1 Schouten tensor f µ a 1 ...an in terms of the frame field e µ a 1 ...an , a relation that involves 2s − 2 derivatives. In order to be able to introduce interactions, i.e., a stress tensor on the r.h.s. of all the Cotton equations for arbitrary spin we must relax the equation F = 0 and instead consider F = T where the r.h.s. must have the property that all the field strengths F a 1 ...an (0, 2n) pick up the proper source terms.
Having concluded that all the component equations F (n q , n p ) = 0 for n q > 0 can be solved we note that the field strength F reduces to i.e., it has become a field with the same structure as Φ defined above. This reduction of F may, however, not be compatible with the Bianchi identities. Also other parts of F that are assumed zero here are probably only so in the linear analysis performed in [1]. As will be elaborated upon elsewhere [19] some of the constraint equations of the spin 3 system along other generators than K ab will contain the spin 2 Cotton tensor and will thus be affected by the introduction of source terms. Hence this description of the structure of F implies that F can not be made to act directly on the vacuum like in the unfolded equation (2.1) since then we would loose information.

Field equations with source terms
The comment at the end of the previous subsection suggests that the proper equation for F and T involving the vacuum is instead the integrability equation for (2.1) namely 6 with F taking values in the whole HS algebra, which implies that from which it should be possible to construct T .
We have now explained how to view the two equations (2.1) and (2.2) and also defined the r.h.s. of these equations without giving any explicit expressions for them. The main result of this paper is that it is in fact possible to construct the r.h.s. producing a fully interacting theory with back reactions in both equations. We emphasize here that we have not yet provided a complete proof that the equations we propose constitute a consistent system. In this context it might also be relevant to analyze the equation (where ⋆ is the

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Turning to the source terms we start by constructing S. It was explicitly shown in [1] that the unfolded equation with a zero r.h.s. gives rise to the conformal interaction term Rφ with the correct coefficient for three dimensions. The goal now is to construct a r.h.s. such that also the fifth order interaction term is generated after unfolding the equation. In fact, also the scalar terms in the full Cotton equation (1.3) require the addition of a source term. The only structures that can be written down which are one-forms with dimension L − 1 2 and could generate the wanted terms are in fact, where λ 1 and λ 2 are two free parameters. Here we have used the definitions where M a and K a are the spin 2 Lorentz and special conformal generators of dimension L 0 and L 1 , respectively. Unfolding (2.1) indeed gives the correct Klein-Gordon equation at the spin 2 level (see below) and interestingly enough also the correct Cotton equation. As described for spin 3 in [1] this unfolding can be carried out further up in spin without any problems. There are, however, features involving infinite sets of higher spin terms in the full equations which probably means that the equations have to be iterated and truncated at some desired high spin level. 7 However, for this to work in the sense of producing the spin 2 Cotton equation with the correctly coupled scalar field as in (1.3) one further step is required. As we will clear below we have to make use of the possibility to shift the gauge fields in A by tensor terms which for spin 2 we choose asÂ (2.24) We will, however, not work with this shifted gauge field but instead move the tensor term over to the r.h.s. of the unfolded equation. Combining this term with the one already in S we find that the r.h.s. becomes Note that a corresponding term for the P a generator does not exist since the term P in A is of dimension zero so it cannot contain any factors of Φ. It may be mentioned in this context that the unfolded equation will itself produce the full spin 2 Cotton equation with the stress tensor as a source. For higher spins one may speculate about the structure of the corresponding Cotton equations. For spin 3 for instance, the Cotton tensor is in the irrep 7 and has dimension L −4 , compared to L −3 for spin 2, and hence the two-scalar terms must contain one further derivative which should result from the unfolding. Also other more complicated terms are possible with derivatives distributed between scalars and JHEP08(2016)142 higher spin frame fields in various ways. It is even possible that there is a non-derivative e µab φ 10 term as a source in the spin 3 Cotton equation which may come from further terms in the source S. E.g., one may envisage terms containing e µ ab which must involve e µ ab M ab , e µ ab K ab , etc, multiplied by |Φ| 4 Φ, |Φ| 8 Φ, etc for dimensional reasons. How it is possible for such terms in S to affect the spin three equations will be clear below. Note that the issue of whether or not terms like these will contribute also to the Klein-Gordon equation depends on traces like e µ a e µ ab being non-zero. However, this is probably not the case since they can be set to zero by higher spin "scale" transformations.
In a similar manner we may deduce the structure of T in the HS equation (2.2). T must be a two-form of zero dimension giving rise to, after unfolding, both ∂ µ φ∂ ν φ and φD µ ∂ ν φ type terms. An especially intriguing fact is that derivative terms of the kind φD µ ∂ ν φ may only arise through unfolding. The T that has these properties will not be presented here and we hope to come back to this question elsewhere. Note that a structure similar to S, i.e., where the two-forms are will not suffice since terms with explicit derivatives seems to be needed. In fact, this follows directly from (2.20) which of course will imply relations between parameters in T and S. Nevertheless, it is the first term in (2.27) that has the correct structure to generate the required source term for the spin 2 Cotton equation in F = T . As for S also T will contain HS contributions of the kind e µ ab P ab etc.

Explicit unfolding
In order to perform some checks we need to unfold the scalar equation

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where we need to keep in mind that the uncontracted flat indices are always in irreps, i.e., in symmetrized traceless representations. This means that for levels n ≥ 1 each equation splits into three irreducible parts n − 1, n and n + 1 obtained by multiplying it with the level one generators P µ := e µ a P a , M µ := e µ a M a and K µ := e µ a K a , respectively. We refer to the resulting equations as n − , n 0 , n + , respectively. Applying this procedure to the n = 1 equation above we find Setting DΦ|0 q = 0 we can insert the level n = 0 result into these equations and find At level 2 we find the following l.h.s. of the unfolded equation To get a feeling for the non-trivial information in these equations we again assume DΦ|0 q = 0 and continue by analyzing the first of the n = 2 equations, the 2 − one. To do that we need to use information from the two lower levels. This gives (dropping s ≥ 3 terms) Then using the fact (where the zero torsion condition is assumed) and the Klein-Gordon equation, we find the above 2 − equation to read We note then that this equation becomes an identity if we set

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where S µν is the Schouten tensor. As we have seen above in (2.16), the equation F = 0 also contains this information. The equation n = 1 + above can now be seen to play an interesting role in rewriting the Cotton equation in a way that will help us to guess source terms for the entire higher spin system. The point is that if we use (2.46) in the n = 1 + equation we can eliminate the term D (µ φ ν) = −D µ ∂ ν φ from the Cotton equation in (1.3). This gives Using the Ricci identity this equation reads and setting f µν = 1 2 S µν as found above it reduces to where C ab = ǫ µν(a D µ R ν b) is the Cotton tensor. Again we find information present also in the equation F = 0. Thus is it clear that while the F = 0 equations contain, of course, only higher spin dynamics without scalar field sources the unfolded equation DΦ|0 q = 0 contains dynamical information for both the scalar field and the higher spin fields but without any non-trivial couplings between the scalar field and the higher spin ones. Introducing sources must thus be done for both equations in a consistent way. We will address this issue again below. We now introduce the non-zero source terms (2.25): One crucial property of this expression for the source is that it is zero at level n = 0, and that S M contributes only to the n 0 equations at level n while S K contributes only to the n − equations. Thus there are no source terms affecting the n + equations which therefore are the same as for DΦ|0 q = 0 where it is used to determine the fields φ a 1 ...a n+1 in terms of fields at lower levels. This is seen as follows: consider a general term at level n in the expansion of K µ Φ 5 |0 q = e µ a K a Φ 5 |0 q which we write as e µ a (Φ 5 ) b 1 ...bn K ab 1 ...b n−1 |0 q . Contracting it with P µ then gives K b 1 ...b n−1 . Contraction with M µ gives instead e µ a ǫ µ

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We will now continue to analyze the effects of adding the explicit source terms given in (2.25): In the case of S K we need the results We find the following contributions to the n − equations P a 6λK a Φ 5 |0 q = 0, n = 0, (2.61) With these results for the source terms of the unfolded equation the first few level equations become, dropping terms involving higher spin fields, (2.72)

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where we have adopted the solution f µν = 1 2 S µν = 1 2 (R µν − 1 4 g µν R) to the n = 2 − equation in order to obtain the Cotton equation from the n = 2 0 equation. The Cotton equation obtained this way has the same structure as the one we were seeking namely the one given in (2.47) so this seems to be on the right track. A crucial further test is to construct the source T in the adjoint equation F = T such that the same Cotton equation results as discussed above after equation (2.27). This should be possible to do and we hope to come back to this in a future publication. 8 We should also emphasize another feature of the calculation leading to these results. The fact that the scalar self-interaction term K|Φ| 4 Φ gives rise to new terms in both the 1 − and 2 − equations leads to a consistency check in the sense that these terms are seen to cancel in the the 2 − equation and hence the result quoted for f µν is not affected by the addition of the K|Φ| 4 Φ term. 9

A cascading Lagrangian
In this section we will make use of a feature of the Chern-Simons gauge theory for the higher spin gauge field A that allows us to show that the component Lagrangian is naturally expressed in terms of the generalized spin connections as suggested in [1]. We will demonstrate explicitly that it is possible to derive such a Lagrangian once a certain subset of the F = 0 component equations are solved. This subset of equations, which will be called cascading, does not include the spin s Cotton equations which are therefore obtained by varying the resulting Lagrangian with respect to the frame fields for each spin s ≥ 2. F = 0 contains a number of other equations that one must use to determine other fields contained in A or prove are identities; for a complete discussion of the spin three situation see [1].
We use here the results coming from the single commutator terms in the spin 2 and 3 cases as examples of the technique. However, these examples make it plausible that this procedure works for all spins and in the full star product formulation where all multicommutators and non-linearities are included. Its origin is in the gauge Chern-Simons theory where the trace is in the higher spin algebra which should generate precisely the terms in the Lagrangian used in the cascading procedure described below. This Lagrangian can in principle be written out explicitly in terms of all the fields appearing in A [7]. This will 8 In order to ensure full higher spin symmetry the source terms S and T (see the beginning of this section) should perhaps be constructed together in an iterative fashion similar to the procedure used in [17] (see also [18]) for non-conformal unfolded systems. However, the conformal symmetry present in the system analyzed here probably means that both the procedure and the result will be rather different. We are grateful to E.D. Skvortsov for discussions on this issue and for emphasizing the possible role of the approach presented in [17]. 9 Concerning the parameter λ2 of this interaction term it will most likely stay independent of λ1 after imposing full HS symmetry (see previous footnote). As mentioned in the Introduction these interaction terms are independent in any conformal spin 2 -spin 0 system without supersymmetry and the HS symmetry will probably not change this situation. There will probably be no other free scalar interaction parameters after implementing HS symmetry but it can not be ruled out at this point.

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produce very complicated expressions and seems useful only for lower spin truncations. To get from this first order formulation to a "second" order one in terms of only the frame fields seems even more complicated and any kind of simplifications that can be utilized in this context would be welcome. Below we will describe one potentially useful feature of this kind.
The standard spin 2 Chern-Simons like action reads in terms of ω 1 := ω(1, 1) which leads to the field equation C µν = 0 for the Cotton tensor C µν = ǫ µ αβ D α (R βν − 1 4 g βν R). Here we use the notation from [1] for the s = 2 spin connection obtained by solving the zero torsion condition. In that paper it was suggested that this spin 2 action has a generalization to arbitrary spin in the sense that the action is naturally expressed in terms ofω n :=ω(n, n) which is a one-form in the irrep n of SO(1, 2). (A spin 3 example is ω ab appearing in the expansion of A 2 in (2.5).) Here we will show how to derive the action for spin 3 which is of the suggested form Note that a cubic term with threeω 2 does not exist. However, once the multi-commutators are taken into account there will appear new interaction terms containing spin connections with arbitrarily high spin. Also other higher spin fields will occur in these interaction terms and to be clear about which fields we talk about we consider again the spin 3 higher spin gauge field in (2.5) A 2 = e ab P ab +ẽ abP ab +ẽ aP a +ω abM ab +ω bM a +bD +f aK a +f abK ab + f ab K ab . (3.4) Solving the first four of the equations in (2.17) will result in a cascading sequence of relations that will express the one-form field f ab , called the spin 3 Schouten tensor, in terms of the frame field e ab each step producing a new derivative. The last equation in (2.17) is then the spin 3 Cotton equation containing five derivatives. The action we derive here uses only the fieldω ab in A 2 denotedω 2 in (3.3). In fact, as we will see below also the fieldb in (2.5) will appear in the action but it turns out that this field is expressible in terms ofω 2 as shown in [1]. The main goal of this section is to give a simple procedure for deriving a Lagrangian that gives the full non-linear Cotton equation for any spin, and indeed for the whole higher spin system. This result follows provided some basic conditions to be specified below are met. Here we give the main ideas and the details only for spin 2 and 3 but it is likely that this method can be generalized to the whole higher spin theory.
We will now show how one can derive an action that automatically gives rise to the fifth order Cotton equation for this spin 3 system. In order to streamline the discussion we simplify the spin 3 equations in (2.17) as far as possible without destroying features of the system that are relevant for this particular discussion. First we note that the spin 3 equations contain fields from the spin 2 system that we can discard at this point but JHEP08(2016)142 put back if a complete analysis is required. This statement applies to all terms containing the spin 2 fields ω a and f a but not to the terms with a dreibein e a . This reduces the equations to We now need to make use of the possibility to gauge fix the higher spin symmetries to further simplify these equations. As explained in [1] the fieldẽ a can be set to zero by using the symmetries related to the parametersΛ ab (2, 2),Λ a (2, 2), andΛ (2,2). This sets to zero the last term in the first equation above. In order to eliminate also the last term in the second and third equations we need to be able to choose a gauge whereω a = e aω andf a = e af . However, while this is possible forω a it is not so forf a . In the former case we can useΛ ab (1, 3) andΛ a (1, 3) to establish this fact but in the latter case we have onlỹ Λ ab (0, 4) at our disposal which means that the best we can do is to gauge fix tõ which unfortunately will complicate the situation somewhat. Instead of trying to construct a Lagrangian directly for the frame fields and then perform a variation with respect to the frame fields to obtain the Cotton equations these can be obtained in a manner that is slightly easier if we make use of the description of this system as a Chern-Simons gauge theory for the conformal group SO (3,2). In order to see how this is done we consider first the spin 2 Chern-Simons system which is given in terms of the gauge field A 1 = e a P a + ω a M a + bD + f a K a , (3.11) where the SO(3, 2) generators P a , M a , D, K a have been assigned one gauge field each. The exercise is then to solve the zero field strength equation F 1 = 0 which if decomposed along the different generators become (here the Riemann tensor is R a = dω a + 1 2 ǫ a bc ω b ∧ ω c and we have imposed the gauge b = 0) which we call the cascading equations while the remaining equation is a constraint on the solution of the cascading system above. The first equation is solved for the spin connection in terms of the dreibein, the second for f a in terms of the Riemann

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tensor with the result that f a µ e νa is just the symmetric Schouten tensor. The last equation is then a constraint that is automatically satisfied while the last of the cascading equations becomes the Cotton equation. The goal is now to use these cascading equations to show that the variation of the action gives the Cotton equation.
We start the cascading procedure from (3.14) The variation of L 1 is which would give the Cotton equation by demanding δL 1 = 0 if the last two terms could be gotten rid off. To achieve this we note first that the second term vanishes due to the torsion constraint after an integration by parts. To deal with the last term we add the standard Chern-Simons term where we have used the second equation in (3.12) in the last equality. Thus we obtain the Cotton equation as a result of varying the Lagrangian L = L 1 + L 2 . However, L 1 = 0 after an integration by parts as a consequence of the torsion constraint which is assumed solved in this analysis. This implies that the Lagrangian L 2 alone provides the Cotton equation when varied with respect to the dreibein e µ a . This derivation of the Cotton equation is a bit too trivial to be interesting but for spin 3 and higher it seems to simplify the calculation of the spin s Cotton equation quite a bit. Recall that these equations are of order 2s − 1 in derivatives. We now turn to the spin 3 system and repeat these steps. To this end we note that the spin 3 Cotton equation in the simplified version given above follows trivially by varying the Lagrangian L 1 = e ab ∧ df ab , (3.18) with respect to the explicit frame field. However, this conclusion is only correct if we can eliminate the second term in its variation But this can be done by adding another term to the Lagrangian whose variation cancels the last unwanted term. The term we need to add is The reason this works is that in the variation

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the first term equals −de ab ∧ δf ab by using the field equation for e ab coming from F = 0 (recall that we are in a gauge whereẽ a = 0). To make use of this field equation is of course allowed here since it is algebraic and actually solved so that it is identically satisfied. This fact can now be used for all the "field equations" in F = 0 except the last one which is the five derivative Cotton equation.
Having established this cancelation we now need to add a further term to cancel also the second term in δL 2 above. The required term is whose variation can be written, again making use of the algebraic "field equations" this time the ones forf ab andẽ ab , as The remaining term is then the last one in the previous equation which we cancel by adding which varies into After canceling the last term against the same term coming from δL 3 we are left with the first term in δL 4 which we write as − 1 6 δω ab ∧ (dω ab − e a ∧f b ). (3.26) The first term in this expression is canceled by the variation of Now we can use the algebraic equations from F = 0 again to find that L 1 + L 2 = 0 and L 3 + L 4 = 0. Since the procedure stops here L 5 is actually (the main part of) the final answer and is precisely the Lagrangian proposed in [1]. We have, however, still one term that we need to deal with, namely the second term in (3.26), which as we will now see is of a slightly different nature. We start by adding Thus the last spin 3 term to add is We have therefore shown that the spin 3 part of the Lagrangian reads The second term on the r.h.s. is related (see above) to the last one and we find the final form of the Lagrangian to be which is therefore expressed entirely in terms of the spin connectionω ab of the spin 3 sector.
One also needs to verify that the remaining constraint equations are satisfied as explained in [1]. It would then be interesting to see how the different steps in the cascading procedure are affected by increasing the spin, adding non-linear terms and coupling the system to other fields. For the spin 2 -spin 3 system these questions may be answered by the analysis of the full non-linear equations in [19]. As noted previously in this section the cascading trick is suggested by writing out the original Chern-Simons action for A using the trace over the higher spin algebra at each spin level separately as done in [7].

Conclusions
In this paper we have continued the approach to conformal higher spin theories in three dimensions set up recently in [1]. There it was emphasized that the unfolded equation DΦ|0 q = 0 for the higher spin algebra based on SO (3,2), the conformal group in three dimensional space-time, realized in terms of two hermitian spinor operators q α , p α satisfying [q α , p β ] = iδ α β , produces the correct Klein-Gordon equation for a conformal scalar coupled to the spin 2 metric and its generalization to spin 3. 10 Here we take this approach some steps further by proposing an unfolded equation for a scalar field coupled to all higher spins ≥ 2 including the φ 5 self-interaction term in the Klein-Gordon equation. The expected scalar interactions with the spin 2 and higher spin fields show also up in the field equations and are produced in the unfolding. That the correct spin 2 Cotton equation is obtained is checked explicitly while for spin 3 the corresponding equation can easily be derived from this setup but it needs to be checked independently. Such a check would strengthen the argumentation for the higher spin equations suggested here.

JHEP08(2016)142
We also present a simple method by which the Lagrangian for each higher spin field can be derived starting from the F = 0 equation valued in the higher spin algebra. This is demonstrated in section 3 for a truncated version of the spin 2 and 3 equations and can be seen to be a consequence of expanding the Chern-Simons gauge theory action using the trace over the entire higher spin algebra. Then a cascading trick leaves the whole action written in a form where the spin connections for each spin play a central role. The spin connections are here expressed as s − 1 derivatives acting on the spin s frame field implying that L n = ω n dω n for s = n + 1 contains 2s − 1 derivatives as it should. The result is a "second" order formalism type Chern-Simons Lagrangian generalizing the standard one L 1 (ω 1 (e)) for spin 2 to all spins. The cascading is here only performed for the linearized theory but will most likely give the full answer once all interaction terms are included.
This approach also suggests a way to write down a higher spin Lagrangian in the higher spin language. One may try to combine the gauge Chern-Simons action S = 1 2 T r (AdA + where we have introduced a dual scalar fieldΦ(q) which is expanded in terms of even powers of q α instead of p α as for the ordinary scalar field Φ(p). By assumption the dual fieldΦ(q) = φ(x) +φ a (x)P a + . . . where P a = − 1 2 (σ a ) αβ q α q β is non-zero on the vacuum p 0| which is used to produce a well-defined inner product p 0|0 q = 1.
This action can be expanded in component fields whose field equations should correspond to the scalar field equation D ⋆ DΦ(p)|0 q = 0. The source S can probably be hidden in the covariant derivative. The unfolded equation DΦ(p)|0 q = 0 could then be regarded as a solution to this equation which means that some information is lost if one instead solves only D ⋆ DΦ(p)|0 q = 0. It would be nice to have an action principle that directly generates the unfolded equation as the field equation. Interaction terms for the scalar field may also arise by considering actions of the kind S 6 = p 0|(Φ * (q)) m ⋆ (Φ(p)) n |0 q | m+n=6 . (4. 2) The actions considered here are dimensionless and the integrands are three-forms but they seem not to produce in a simple way the field equations used previously in this paper. The main reason for this is that although the dual fieldΦ(q) is here assumed to start with φ(x) the terms of higher order in q α will probably be very complicated (even non-local).
Another potentially interesting aspect arises if this higher spin theory can be generalized to contain the topologically gauged spin theories derived in [2,3,20]. Then perhaps the background solutions found in [20,21] could be lifted to the higher spin theory which could then provide information about how to write, e.g., an action also in AdS 3 , Schroedinger and the semi-flat Schroedinger geometries discussed in [21,22]. If this turns out to work it would give additional support for a "sequential AdS/CF T " phenomenon as suggested in [23] where Neumann boundary conditions and the associated dynamical conformal boundary theories play a crucial role.