Cubic interactions of massless bosonic fields in three dimensions II: Parity-odd and Chern-Simons vertices

This work completes the classification of the cubic vertices for arbitrary spin massless bosons in three dimensions started in a previous companion paper by constructing parity-odd vertices. Similarly to the parity-even case, there is a unique parity-odd vertex for any given triple $s_1\geq s_2\geq s_3\geq 2$ of massless bosons if the triangle inequalities are satisfied ($s_1<s_2+s_3$) and none otherwise. These vertices involve two (three) derivatives for odd (even) values of the sum $s_1+s_2+s_3$. A non-trivial relation between parity-even and parity-odd vertices is found. Similarly to the parity-even case, the scalar and Maxwell matter can couple to higher spins through current couplings with higher derivatives. We comment on possible lessons for 2d CFT. We also derive both parity-even and parity-odd vertices with Chern-Simons fields and comment on the analogous classification in two dimensions.

The covariant classification of cubic vertices in [22] not only completed the light-cone vertices of Metsaev [11] to off-shell ones for Fronsdal fields but also defined a scheme of field redefinitions in a given cubic action to bring it to the form containing not more than s 1`s2`s3 derivatives. This form does not contain any contraction between derivatives and is uniquely defined. We refer to it as a vertex in Metsaev basis. This was implemented later in [43,44] for translating the quadratic order of the Vasiliev equations in pAqdS 4 , corresponding to cubic action, to the Metsaev basis in metric formulation, that is, AdS extensions of Minkowski vertices for each number of derivatives ∆ " s 1`s2`s3´2 n for n " 0, 1, . . . , mints 1 , s 2 , s 3 u.
Attempts for going beyond cubic order [23,[45][46][47][48][49] have met difficulties in the framework of local field theory. An interesting suggestion for a possibility of a non-local theory with conformal symmetry has been made in [49] which calls for further studies.
Another interesting recent development is the progress in the holographic reconstruction [50][51][52] of type A HS theory in AdS d`1 . Together with the aforementioned attempts of construction of a quartic order action via the Noether procedure, these results brought to the forefront of HS research the puzzle of locality which, to our best knowledge, was first posed sharply for three dimensional systems in [3]. One may hope that the key to the solution of this puzzle can be found more easily in the three-dimensional case by applying recently obtained knowledge of the metric-like theory. Unfortunately, most of the aforementioned advances in higher-dimensional HS gravities are not directly applicable to three dimensional models. This is due to the heavy use of Metsaev basis of cubic interactions in higher dimensions that does not apply to d " 3. In order to make use of new results in metric-like HS gravity also for the three-dimensional models, one first needs to address the gap in the classification of cubic vertices. In this paper, we continue the study aimed at filling this gap initiated in [1] where parity-even cubic vertices of massless bosons were classified. We complete the three-dimensional classification of cubic interactions deriving parity-odd vertices for massless bosonic fields as well as their couplings to Chern-Simons fields. We also elaborate on the analogous classification in two dimensions in the Appendix.
Despite all the successes of the three-dimensional HS gravities (see [3,[53][54][55][56] and references therein), there is no action formulation 2 for the only known example of higher spin theory with propagating degrees of freedom in three dimensions i.e. Prokushkin-Vasiliev theory [3]. This theory contains scalar degrees of freedom interacting with higher spin gauge fields which do not carry propagating degrees of freedom in the bulk. The Chern-Simons formulation of HS gravity in three dimensions does not answer the question whether a Lagrangian for the Prokushkin-Vasiliev theory exists. This question may be tackled in the metric-like formulation where scalar and gauge fields can be put into interaction in a straightforward manner. This approach is much less explored in three dimensions though, with the exception of a few works on higher spins in the Fronsdal formulation [58][59][60].
S-matrix methods do not apply to three dimensions where massless particles of spin s ě 2 do not propagate. For the same reason, Metsaev's light-cone classification [11] does not work in three dimensions. The part of the cubic vertices that contains no divergences and traces i.e. traceless-transverse (TT) vertices are non-trivial though, as shown in [1], and can serve as the basis for the classification of cubic interactions of massless fields in three dimensions. The main difference between dimensions d ě 5 and d ď 4 for the cubic interactions of massless symmetric HS fields is the existence of dimension-dependent identities (Schouten identities) that are available in the latter case. Due to these identities, the classification of cubic vertices in three dimensions becomes a completely independent problem which overlaps with the generic dimensional classification only for some vertices involving lower spin fields. The classification of d " 3 vertices was initiated in [1] where the parityeven vertices for interactions of massless bosons were derived. In this work, we complete the classification adding to it the parity-odd vertices of massless bosons as well as their interactions with Chern-Simons vector fields.
The paper is organized as follows: In Section 2 we review metric-like formulation of free HS fields. In Section 3 we review the construction of cubic vertices in higher dimensions, and the parity-even vertices in three dimensions. In Section 4 we derive full list of parityodd cubic vertices of massless bosons in three dimensions and establish interesting relation between parity-odd and parity-even vertices. In Section 5 we study interactions of massless fields with Chern-Simons vector fields. We conclude by summary of results and discussion in Section 6. Appendices provide curious observations related to the parity-even vertices and classification of Fronsdal cubic vertices in two dimensions.

Review: Free Theory
In this paper we study interactions of massless fields of any spin as deformations of the free theory. To this end, we first set the stage by describing the free theory. In order to streamline the notation, we will contract spacetime indices µ, ν, . . . with commuting auxiliary variables a µ . In this language, the rank s symmetric tensor field is given by: In order to describe a free particle with spin s in a covariant manner, one has to impose on the rank s symmetric Lorenz tensor field the so-called Fierz equations [61]: For the massless fields pm 2 " 0q, one has to require also an extra equivalence between fields, differing by a gradient shift with traceless and transverse parameter ǫ s´1 px; aq: δφ s paq " pa¨B x qǫ s´1 paq , pB x¨Ba qǫ s´1 paq " 0 , B 2 a ǫ s´1 paq " 0 . 3) It has been a challenge to find a Lagrangian, even for the free Fierz equations. The natural expectations based on experience with lower spins is to have a single equation of motion for the rank s tensor field, which has all the three Fierz equations as its consequences and also gauge symmetry of action in the massless case. For the massless case, the most conventional description is due to Fronsdal [8]. The equation of motion is given by Fronsdal tensor: with the de Donder operator Dpaq " pB x¨Ba q´1 2 pa¨B x qB 2 a . The Fronsdal tensor F is invariant with respect to gauge transformations: The Fronsdal field φ s paq is doubly traceless: The action is given by: with Lagrangian equations of motion: Using double-tracelessness of the Fronsdal field, one can easily show that the equations of motion G " 0 are equivalent to the Fronsdal equations F " 0. At the linearised level, the Fronsdal equations imply the Fierz equations. An alternative to the Fronsdal action is given by the Maxwell-like formulation [62] of HS dynamics. The traceless-transverse (TT) parts of vertices in both formulations are equivalent though [29] and we will therefore not distinguish them in this work since we restrict ourselves to TT vertices only following [1]. The TT vertices studied here can be completed to off-shell vertices for both Fronsdal and Maxwell-like HS fields 3 . One can regard the results of this work as the classification of deformations of the Fierz system of equations [61] for massless HS fields in d " 3. There is an important difference between Fronsdal and Maxwell-like descriptions relevant to this work which we do not elaborate on here, though. While Fronsdal fields do not carry propagating degrees of freedom in three dimensions, the reducible Maxwell-like fields do carry a propagating massless scalar (vector) degree of freedom for even (odd) spin. As a consequence, non-linear theories of Maxwell-like fields, if any, cannot be given by Chern-Simons actions in striking difference with many known models for Fronsdal fields. The classification that we carry out here can be implemented for building models with both Fronsdal and Maxwell-like field content.

Review: Cubic Vertices
We will assume that there exists a gauge invariant non-linear action S that can be expanded in power of fields with a small expansion parameter g as follows S " S p2q`g S p3q`g2 S p4q`. . . , (3.1) where S p2q " S ps 1 q`Sps 2 q`Sps 3 q with S ps i q denoting the Fronsdal action for the spin s i field (2.7). Gauge invariance of the action implies Using the fact that δ p1q S p2q " δ p1q φ G, it follows that 4 where « denotes equality upon imposing free equations of motion G " 0. Note also that any two actions S and S 1 related by a field redefinition φ Ñ φ`g f pφ, φq, obey This ambiguity in field redefinition at the cubic order will be fixed by restricting the possibility of derivative contractions in the cubic vertex (as reviewed for example in [1,33]).
One can now make the following ansatz for the cubic action where the differential operator V " VpB x 1 , B a 1 , B x 2 , B a 2 , B x 3 , B a 3 q is to be determined. Since V is a scalar operator, it is built of contractions of the derivatives B a i and B x i . It can be shown that, up to total derivatives and upon fixing the freedom in field redefinitions 5 , all contractions can be written in terms of Let us furthermore restrict to interaction terms which do not involve traces T i or divergences Div i . In this case, one obviously has V " Vpy i , z i q .
(3.6) 4 Note that our notation is somewhat schematic. The variation is to be understood as a sum of three terms, i.e. δ p1q φ G " δ p1q φ ps 1 q G ps 1 q`δp1q φ ps 2 q G ps 2 q`δp1q φ ps 3 q G ps 3 q . 5 The field redefinition freedom is fixed following [21,22], that is, by removing all terms with derivatives contracted with each other. Strictly speaking, one can exclude the derivative contractions by field redefinitions only in the terms that do not contain divergences. That turns out to be already sufficient for fixing the field redefinition freedom (see, e.g., [25]).
The gauge variation of the ansatz for the cubic action is then given by where here and in the following we assume indices i, j, . . . to be cyclic in p1, 2, 3q, for example y i " y i`3 . Using the commutators where  denotes equality up to equations of motion, total derivatives, traces and divergences. Similarly, it can be shown that all other commutators vanish up to these terms. After dropping total derivatives with respect to B x i , it then follows that It then immediately follows that gauge invariant vertices solve the equations 8) and are given by In generic spacetime dimension, these solutions span the entire space of possible cubic vertices. However at fixed dimension d ď 4, most of these vertices are vanishing [33], while there may be more solutions due to Schouten identities as demonstrated in [1]. We will briefly review the main results of the latter work on parity-even vertices in d " 3 here.
3.1 Parity-even Vertices in d " 3 The derivation of Lorentz covariant cubic vertices described above has to be supplemented with Schouten identities that are relevant for d ď 4. The case of four dimensions can be found in [33] while in [1] the d " 3 parity even vertices were classified. The Schouten identities can be systematically derived by "over-antisymmetrisation" of Lorentz indices and there are even Mathematica packages doing so [63]. The elementary three-dimensional Schouten identities for parity-even TT cubic vertices are given as (grouped in two-, threeand four-derivative identities, no summation over repeating indices assumed): These identities will be supplemented with parity-odd ones in the next section and are needed for the derivation of parity-odd vertices. Due to identities (3.10), the classification of parity-even cubic vertices in 3d is different from that of d ě 4. In particular, these identities allow for existence of two-derivative and three-derivative TT vertices given by [1]: The expressions (3.11a) and (3.11b) describe unique cubic vertices for even and odd sum of spins respectively. Note that (3.11a) involves minimal coupling to gravity discussed for particular cases earlier in [58,64,65]. These vertices exist only if the spin values satisfy triangle inequalities s i ă s i`1`si´1 .

Parity-odd Vertices for Massless Bosons
In order to construct parity-odd vertices of massless fields in three dimensions, one needs to add to the building blocks of the parity-even vertices, i.e. y i and z i , all elementary scalar contraction operators that involve the invariant tensor ǫ µνλ of the Lorentz algebra. These are: where the V 's satisfy (discarding total derivative terms) while the W 's are a choice of basis for nine different structures with two derivatives related to each other up to total derivatives. Therefore the independent set of parity-odd variables is spanned by ten scalar operators U, V ij pi ‰ jq, W i . It is straightforward to check that: up to total derivatives. The operator D i (3.8) takes the following form for parity-odd vertices: The elementary parity-odd Schouten identities are given by (with arbitrary i, j, k and no summation over repeating indices assumed): where identities are grouped into one, two and three derivative ones. From these we derive other useful identities: A consequence of the Schouten identities is that all parity-odd terms with more than one derivative can be written in terms of W i , y i , z i operators only as long as the spins satisfy triangle inequalities s i ă s i`1`si´1 . This property will be useful in the following. The terms that cannot be written only in terms of the variables mentioned above are of the following form: We assume without loss of generality s 1 ě s 2 ě s 3 . Then, all the terms of the type (4.12) are given as: and exist only for s 1 ě s 2`s3 . We also note that any expression written in terms of W i and non-vanishing up to identities (4.8c) and (4.11) cannot be converted to V ij expressions that vanish. This is due to the fact that terms involving W k with different k give rise to V ij with different j that cannot sum up to zero through identities (4.8a) and (4.8b). This simple technical observation suggests that working solely with W k -s wherever possible will not miss any information about terms that may conspire to sum up to zero. Since W i -s are also commuting with all gauge variations, it makes them the preferred choice of variables in expressions with more than one derivative that we will study in the following. We now proceed with the derivation of parity-odd cubic vertices for massless bosons in three dimensions. We will need to discuss separately different cases and simplify each ansatz maximally, to save virtual trees that get cut in order to supply us with Mathematica notebooks.

Vertices with Scalars
The simplest example is a vertex with two scalar fields involved: ps, 0, 0q. In this case the only candidate vertex operator is: and defines a gauge invariant vertex of current interaction type: where the currentJ is roughly the curl of the parity-even conserved current J µ 1 ...µs of spin s. Next we look at the possible vertices with s 1 ě s 2 ě 1 and s 3 " 0. The general ansatz for ps 1 , s 2 , 0q vertex can be written as: (4.17) The variation of (4.17) with respect to the gauge symmetry of the spin s 1 field gives: The variation of (4.17) with respect to the symmetry of the spin s 2 field gives: Vanishing of this variation is compatible with a non-zero vertex only for s 2 " 1, α " 0. Therefore, there is a unique vertex: which is invariant with respect to the gauge transformation of the second (Maxwell) field (D 2 V P O s,1,0 " 0), while the gauge variation of the spin s field, vanishes due to (4.8c) iff s ě 2. Therefore, the vertex of type ps, 1, 0q exists for any s ě 2. For s 2 ě 2, the variation (4.19) can be rewritten as: and allows for gauge invariance only for trivial solution α " β " 0. Similar to the parity-even case, there are no couplings of the type ps 1 , s 2 , 0q with s 1 ě s 2 ě 2. Thus we found all the vertices involving scalar fields.

Vertices with Maxwell Fields
From the Schouten identity (4.8c), it follows immediately that there is a vertex of the type ps, s, 1q with two derivatives: It is a parity-odd two-derivative coupling to spin one which requires charged spin-s fields. For s " 1, (4.23) reproduces the spin one vertex found by Anco in [66]. There is another vertex of the type ps, 1, 1q that may be guessed immediately: which involves s`1 derivatives. For s " 1, (4.24) coincides with (4.23) up to relabelling of the fields. We will come back to this vertex shortly. It remains to check other possibilities of interactions s 1 ě s 2 ě s 3 " 1 with Maxwell field. It is straightforward to see that the number of derivatives cannot be less than s 1´s2 simply because there are no candidate scalar expressions. The upper bound on derivatives is a bit more subtle to define. An obvious upper bound is s 1`s2 , since all vertex monomials with s 1`s2`2 derivatives vanish due to (4.8c) for any s 1 , s 2 and there are no candidate expressions with derivatives more than s 1`s2`2 . Nevertheless, it can be easily shown that for s 1 ě s 2 ąą 1 the upper bound is much lower than s 1`s2 due to (3.10c) and (4.8c). In fact, careful examination taking into account all Schouten identities shows that there are no non-trivial vertex candidates for the number of derivatives more than s 1´s2`2 . Therefore we are left with two candidate values for number of derivatives in the vertex: s 1´s2 and s 1´s2`2 . We will consider these cases separately.
ps 1´s2`2 q´derivative vertex. With the help of some elementary algebra and making use of Schouten identities, a general ansatz for an s 1´s2`2 derivative vertex can be written in the form: where γ i are arbitrary constants. For simplicity, we discuss separately the cases of s 2 " 1, s 1 " s 2 and s 1 ą s 2 ě 2.
• For s 2 " 1, we have a general ansatz For s " 1, we have: with arbitrary γ i . Each of the three terms in this expression is separately gauge invariant and defines a vertex of the type (4.23). We have three inequivalent vertices, defined for any triple of Maxwell fields. As opposed to the Yang-Mills vertex, which is fully antisymmetric in all the three fields involved, the term W i z i is antisymmetric only in two fields A i˘1 µ and can even define a cubic vertex for only two distinct Maxwell fields (e.g. taking value in the two-generator Lie algebra of infinitesimal affine transformations of a real line). This vertex has been studied in [66]. One can write it in explicit form: For s ě 2, we have: This is the ps, 1, 1q vertex (4.24) where, for odd s, non-trivial interaction requires charged Maxwell fields.
• For s 1 ą s 2 ě 2, the general ansatz (4.25) reduces to: which is not gauge invariant under the variation of the second field with spin s 2 .
ps 1´s2 q´derivative vertex. A general ansatz with s 1´s2 derivatives can be written for s 1 " s 2 " s in the form (without derivatives since s 1´s2 " 0): (4.32) It is elementary to check that this expression is not gauge invariant. We will assume in the following that s 1 ą s 2 , in which case the general ansatz takes the form: and it is easy to check that the equation D 1 V s 1 ,s 2 ,1 " 0 have only vanishing solutions for the coefficients α, β, γ unless s 2 " 1, β " 0, γ "´α. The only candidate expression V s,1,1 " pV 23 z 3´V32 z 2 qy s´2 1 is however not invariant with respect to the gauge transformations of Maxwell fields.
We conclude that all the parity-odd vertices with Maxwell fields are given by (4.23), (4.24) and (4.27).

Gravitational Interactions
Making use of the Schouten identity (4.8c), one can easily show that there is a threederivative parity-odd ps, s, 2q coupling to massless spin two: This vertex is symmetric with respect to the exchange of spin s fields and therefore does not require charged fields. For s " 2, the expression (4.34) reproduces the vertex found by Boulanger and Gualtieri in [67].
Any ps, s, 2q type of parity-odd vertex requires an odd number of derivatives. We will see in the following that there are no parity-odd vertices with one derivative or with more than four derivatives. Therefore, (4.34) is the unique parity-odd coupling to gravity for given spin s.
We conclude that the parity-odd minimal coupling to gravity is given by a threederivative vertex. Let us recall that the parity-even gravitational coupling has two derivatives. This is in contrast to spin one (Maxwell) minimal couplings, where the parity-odd coupling (4.23) has two derivatives while the parity-even coupling has three derivatives [1].
It remains to see what the other options of s 1 ą s 2 ě s 3 " 2 couplings are. These vertices will be classified in the following where we will consider the more general case of couplings between fields with arbitrary spin values.

General Case
It is elementary to verify, by making use of Schouten identity (4.8c), that the expression W 1 y n 1 z m 1 is gauge invariant for any n, m and therefore forms a vertex. These type of vertices are all exhausted by (4.14), (4.23), (4.24) and (4.34). There are no vertices of the aforementioned type with n, m ě 2 due to (4.11).
After the examples with low spins, we now start studying more general cases of cubic interactions.

Couplings without Derivatives
It is straightforward to show that there are no vertices without derivatives. In order to do so, one just needs to gauge variate the most general ansatz, and compare to the linear combination of one-derivative Schouten identities with arbitrary coefficients. One will thus verify that there is no such linear combination and therefore no vertex without derivatives.

One-derivative Vertices
It is straightforward to show that any one-derivative parity-odd vertex with three massless fields of spins s 1 ě s 2 ě s 3 ě 2 could be written in the following form: which can be further simplified in case if the spins satisfy triangle inequalities s 1 ă s 2`s3 to and, if they saturate triangle inequality s 1 " s 2`s3 , to V s 1 ,s 2 ,s 3 " pαV 23 z 3`β V 32 z 2 qz n 2 2 z n 3 3 . (4.38) In both cases, even though there are solutions for each of the equations D i V s 1 ,s 2 ,s 3 " 0, there is no non-zero intersection between these solutions 6 . It is also easy to verify that there are no candidate expressions for one-derivative vertices if s 1 ą s 2`s3 . We conclude that there are no parity-odd vertices with one derivative for any spins s 1 ě s 2 ě s 3 ě 0.

Two-derivative Vertices
Now we turn to studying two derivative parity-odd interactions. This corresponds to odd values of the sum s 1`s2`s3 and therefore the triangle inequality cannot be saturated: s 1 ‰ s 2`s3 . We discuss separately the case when the spins satisfy triangle inequalities and when they do not.
Triangle inequalities are satisfied. Taking into account that for s 1 ě s 2 ě s 3 ě 1 and s 1 ă s 2`s3 , any vertex monomial with two derivatives can be brought to the form where the only parity-odd operators are W i . We end up with a simple ansatz: Making use of (4.8c) and (4.10), one can show that the gauge invariance conditions, " pα n 2´β n 1 q U y 1 y 2 y 3 z n 1´1 1 z n 2´1 2 z n 3 3 " 0 , (4.40c) imply: β n 3´γ n 2 " γ n 1´α n 3 " α n 2´β n 1 " 0 . The solution to these equations fixes the vertex uniquely up to an overall constant 7 : This vertex exists for any triples of spins with odd sum satisfying strict triangle inequalities. For s 1 " s 2 " s 3 " 3, the expression (4.42) reproduces the vertex found by Boulanger, Leclerq and Cnockaert in [68]. To our best knowledge, the latter is the only example known in literature of parity-odd cubic vertices of HS fields in three dimensions. This result is similar to parity-even case, where the two-derivative vertex (3.11a) exists for every triple of spins, with even sum, satisfying strict triangle inequalities. One important difference is that if there are two fields with the same spins in the vertex the parity-even vertex with two derivatives, (3.11a), is symmetric with respect to exchange of these fields while parity-odd one, (4.42), is antisymmetric (we assume at least one of the spins is greater than one). We will come back to the relation of the parity-even and parity-odd vertices in the following.
Triangle inequalities are violated. It is elementary to show that for s 1 ą s 2`s3`1 there are no vertex monomials with two derivatives. The only allowed case is s 1 " s 2`s3`1 , with an ansatz involving expressions of the type (4.13): This expression is invariant with respect to the spin s 1 field's gauge variation, D 1 V " 0, if α s 2`β s 3 " 0, and with respect to the second field's gauge variation, D 2 V " 0, if α s 2 " 0 " β ps 2´1 q , γs 2 " 0, while for the invariance with respect to the third field we get: α ps 3´1 q " 0 " β s 3 , γ s 3 " 0. The only non-trivial solutions for this class of vertices are given by (4.14) with s " 1 and (4.20) with s " 2.
We conclude that, similarly to parity-even vertices with two derivatives, there is only one parity-odd vertex (4.42) with two derivatives for each triple of spins s 1 ě s 2 ě s 3 ě 2 satisfying triangle inequalities s 1 ă s 2`s3 and with odd sum s 1`s2`s3 .

Three-derivative Vertices
For vertices with three derivatives, we consider separately three cases depending on the values of spins. This corresponds to even sum of the spins.
Triangle inequalities are satisfied. In this case, the general ansatz for the vertex is given by (the overall arbitrary coefficient is dropped): V P O s 1 ,s 2 ,s 3 "´W i y i z 2 i z n 1 1 z n 2 2 z n 3 3 " U y 1 y 2 y 3 z n 1 1 z n 2 2 z n 3 3 , s i " n i´1`ni`1`2 . (4.44) Remarkably, this vertex is gauge invariant with respect to all three variations due to (3.10c) and (4.11). This result is similar to the parity-even case (3.11), where every triple of spins with odd sum defined a unique vertex (3.11b) proportional to y 1 y 2 y 3 8 . One notable difference is that in case if there are two fields with identical spin, due to the factor U , the vertex is symmetric with respect to permutations of these fields, as opposed to the vertex (3.11b), which would be antisymmetric. For s 1 " s 2 " s 3 " 2 the vertex (4.44) reproduces the symmetric d " 3 vertex of [67].
Triangle inequalities are saturated. In this case, there are no non-trivial vertex monomials with three derivatives.
Triangle inequalities are violated. This case allows for non-trivial vertex ansatz iff s 1 " s 2`s3`2 . The most general ansatz is given by: The analysis of this case is similar to the two-derivative one performed above. The only non-trivial solutions have been covered by (4.14) with s " 2 and (4.20) with s " 3.
It is a straightforward algebraic exercise to show that there are no non-trivial vertices with more than three derivatives for s 1 ě s 2 ě s 3 ě 2. This completes the classification of parity-odd cubic vertices of massless bosonic fields in three space-time dimensions.

Relations between Parity-Odd and Parity-Even Vertices
There is a remarkable universality in the formulas of the vertices (3.11a), (4.42), (3.11b) and (4.44). In order to show it, we first notice the following relation (as always in this work, we neglect trace terms): Now, we can formally define the following operator Now, let us shift the integers n i in the (4.42) by half: n i Ñ n i`1 2 . Then the sum of the spins becomes even and the equation (4.42) can be formally rewritten as: where s i " n i`1`ni´1`2 . Here we used the identities: 49) and (3.10a). The equation (4.48) exactly reproduces the parity-even vertex (3.11a) up to an overall constant. It is elementary to show, that the same relation holds between three-derivative parity-odd (4.44) and parity-even (3.11b) vertices. Another curiosity related to parity-even vertices is discussed in Appendix A. This universality in formulas may have a deeper meaning in terms of certain dualities between fields that is yet to be uncovered. For example, it may be related to the Chern-Simons formulation where each HS field has two connections analogous to dreibein and spin connection of gravity. When switching to the Fronsdal formulation, the "spin connection" is solved in terms of the "frame field" and the solution contains one derivative and a Levi-Civita tensor. Replacing one "frame field" with a "spin connection" partner may result in switching the interactions between parity-odd and parity-even ones (it changes the parity and the number of derivatives by one). This is a speculation but can be checked by explicit computations. It is also tempting to speculate about the existence of a more fundamental formulation of any HS theory in terms of spinors, analogous to [2,3], that treat the parityeven and parity-odd vertices on the same footing. We leave more thorough investigations of this aspect to a future work.

Vertices with Chern-Simons Vector Fields
So far we have been studying TT vertices of Fierz-type fields including the Maxwell field for s " 1 that is given by the free Lagrangian: with field equation B µ F µν " 0, i.e. both Fronsdal equation [8] and Maxwell-like HS equation [62] for s " 1. In three dimensions one can also consider Chern-Simons (CS) vector fields with free Lagrangian: and free field equation F µν " 0. It is common to call CS field "spin one" or "vector" field, but one has to be careful to not confuse it with the Maxwell field. We will mostly use the terms CS or Maxwell for the corresponding fields in the following. Since this field appears naturally in the context of HS gravity theories [3,65], we study its interactions with other massless fields for completeness of our analysis. Note that due to the difference in the free-field equations, the equivalence class that is defined for field redefinitions and gauge variations of vertices is different for CS fields. Any term that is proportional to free Maxwell field equations is obviously also on-shell trivial for CS fields. The opposite, however, is not true. The on-shell trivial cubic terms for a CS field (i´th field in the vertex) that are not trivial for Fierz (i.e. Fronsdal and Maxwell-like) fields are given by the expressions: Together with Fierz equations and Schouten identities, these terms define an equivalence class for cubic vertex monomials. Cubic vertices with CS fields should have trivial gauge variations in this class while not being trivial themselves. There are three possible cases depending on the number of CS fields in the cubic vertex.
Vertices with one CS field. A general ansatz for parity-even vertices with one CS field and two Fierz fields with spins s 1 , s 2 can be written in the form: Note that y 2 z 2 term is absent as it can be replaced by y 1 z 1 due to identity (5.3): y 1 z 1ỳ 2 z 2 " 0. This vertex is gauge invariant with respect to all three gauge transformations for α " 0 and is not trivial only for s 1 " s 2 . It therefore defines a unique cubic vertex of two massless fields of spin s and a CS field: It is straightforward to see that this interaction corresponds to minimal coupling to vector gauge field obtained by replacing in the free action of a spin-s field B Ñ ∇ " B`A, i.e. making use of covariant derivatives. This coupling has appeared for example in [65] and is obviously not applicable to Maxwell fields. The free action of a spin s field, supplemented with covariant derivatives, is gauge invariant up to commutators of covariant derivatives which are proportional to the curvature of the vector field. This curvature terms are equations of motions for CS fields and can therefore be compensated by deformations of transformations for CS fields. This is not true for Maxwell fields though. The absence of minimal coupling to the electromagnetic field is known as the Velo-Zwanziger problem [69] and is analogous to the Aragone-Deser problem for minimal coupling to gravity. In three dimensions, the minimal coupling to gravity exists which is related to the fact that the Riemann curvature of gravity is proportional to Einstein equations in three dimensions and therefore the problematic terms can be compensated with deformations of gauge transformations of the metric (analogously to the Rarita-Schwinger coupling that leads to Supergravity). Even though the mechanisms are slightly different for spin one and spin two, in both cases the fact that the curvature tensor is on-shell trivial allows for minimal coupling. The on-shell triviality of the curvature tensor is, on the other hand, related to the absence of dynamical degrees of freedom in the bulk and opens the possibility for CS formulation. Parity-odd CS vertices are also severely restricted. Given that the third field in the vertex is a CS field, it can for example be shown that W 1 y 1 z 1 " 0 " W 2 y 2 z 2 . After some algebra, it is straightforward to show that, for two Fierz fields with spins s 1 " s 2 " s, there is a two-derivative coupling to CS field with the vertex operator given by: We skip the details of the computations here since they are elementary algebraic manipulations by straightforward application of the Schouten identities, Fierz equations and (5.3).
Vertices with two CS fields. In this case, the extra identities include (we assume the second and third fields are CS): Using these equalities, one can easily show that there are no vertices of interactions between a massless field with spin s and two CS fields if s ě 2. Instead, there is a vertex of interactions of a Maxwell field and two CS fields: For this vertex to be non-zero, the two CS fields should be charged. There are no parity-odd vertices with two CS fields and a massless field with spin s.
Vertices with three CS fields. In this case we have: The only non-trivial contraction between the three fields is given by a parity-odd expression, which is gauge invariant and is the well-known interaction term of CS fields. For this interaction to be non-trivial, CS fields should carry non-abelian charges. These results fit into the picture of our findings for HS fields. There are no cubic interactions between massless fields where spins do not satisfy triangle inequalities.

Discussion
In this work, we completed the program initiated in [1] providing an exhaustive classification of covariant cubic interactions for massless bosonic fields in three dimensions. We found that the parity-odd cubic vertices for interactions of massless fields in three dimensions are in one-to-one correspondence with parity-even vertices. For each collection of massless fields satisfying strict triangle inequalities, there is a unique parity-odd vertex on top of the unique parity-even one 9 . For triplet of spins not satisfying triangle inequalities, the only cubic vertices are of "current-interaction" type, involving two matter fields of spin s " 0 or 1. For the triplets, that contain at least two spins greater than one, all the vertices have either two or three derivatives.
Our results should match the CFT three-point functions, as argued, e.g., in [70]. The uniqueness of the vertex for given triplet is in agreement with the two-dimensional CFT. The three-point functions of quasi-primaries in 2d CFT have two free parameters for every triple of spins. In our classification, for each triple we get one parity-even and one parityodd vertex therefore match the number of independent structures. The only intriguing aspect is the missing vertices, which translate into selection rules in 2d CFT. Similarly to the parity-even case [1], in the parity-odd case, missing vertices are all those containing at least two fields with spin greater than one and violating strict triangle inequalities. Therefore, for quasi-primaries of spin values s 1 ě s 2 ě s 3 ě 2, all the three point functions for values s 1 ě s 2`s3 are expected to be zero. This property is observed in known examples of 2d CFT's (see, e.g. [71]), but we are not aware of a general proof.
The only massless fields that carry propagating degrees of freedom in three dimensions are scalar and Maxwell fields which are related by duality. Nevertheless, there are slight differences in vertices containing scalars and Maxwell fields observed in our classification. When we compare these vertices, one can take into account exact relation of duality between a Maxwell field A µ and a scalar φ, given by the relation: If a vertex involves the curvature of the Maxwell field, one can simply replace it with the right hand side of the equation (6.1) and get a vertex for a scalar, which has opposite parity. Similarly, if the vertex contains derivative of the scalar, one can replace it with the dual of the curvature of the Maxwell field. Instead, for the vertices where one has a naked vector potential A µ this dualization is not applicable since the curl operation is not invertible and the field A µ cannot be expressed through φ locally. Let us start with parity-even vertices with Maxwell fields. There is an ps, 1, 0q vertex, which contains s`1 derivatives, and in which one can dualize the Maxwell field to get parity-odd ps, 0, 0q vertex (4.14) with s`1 derivatives. Alternatively, one can dualize the scalar to get the parity-odd ps, 1, 1q vertex (4.24) with s`1 derivatives. Therefore, we established duality relations: This duality works for any s ě 1.
Next, there is a parity-even vertex V ps,1,1q " y s´1 1 G with s derivatives. Dualization of one Maxwell field leads to a vertex V P O ps,1,0q with s derivatives given by (4.20). We can further dualize the second Maxwell field and get a parity-even vertex V ps,0,0q " y s 1 with s derivatives: These dualities work for s ě 2, since there is no parity-odd vertex for spin configuration p1, 1, 0q, which leaves out the Yang-Mills vertex from dualization procedure. The dualization of the other cubic vertex of three Maxwell fields, F 3 vertex, leads to a trivial TT expression and therefore does not have a parity-odd p1, 1, 0q dual either. The (A)dS cubic vertices in any dimensions can be understood as deformations of flat space cubic vertices and therefore the first step towards (A)dS vertices lies in the classification of their flat counterparts. In fact, all known Lagrangian theories with HS spectrum in three dimensions allow for flat space limit. Therefore, one may expect that the Lagrangian formulation for Prokushkin-Vasiliev theory, if existing, may also allow for a flat limit. Even more, three-dimensional Minkowski vertices can be extended to arbitrary Einstein backgrounds due to the same reason as the absence of Aragone-Deser problem in 3d -the obstructing terms are given by Weyl tensors and therefore vanish in three dimensions. One can even work with full non-linear gravity while constructing the action perturbatively in powers of HS fields (see [65] for such expansions of full non-linear theories). In that case, one needs to take care of the backreaction to the Einstein equations involving HS fields, which contribute to the construction of quartic and higher order vertices.
The full classification of cubic vertices is the first step towards construction of a Lagrangian for the HS theories accommodating propagating degrees of freedom which are not covered by Chern-Simons actions. Our classification is performed for the three-dimensional Minkowski background while the (A)dS extension can be considered straightforwardly. Since the main technical difficulty of (A)dS extensions is related to the non-trivial commutators of covariant derivatives, it is natural to expect that those vertices that contain many derivatives will be the most challenging. As we have seen, in three dimensions the only vertices that contain more than three derivatives are current interactions containing scalar and Maxwell fields. The AdS extensions for these vertices have already been studied in higher dimensions in [19,20,26]. The scalar coupling in three dimensions was studied in [72][73][74][75].
The main technical novelty of the three-dimensional classification provided in this work and in [1] compared to earlier work on cubic vertices in arbitrary dimensions is related to the systematic implementation of Schouten identities in three dimensions. When considering quartic interactions of massless symmetric fields, there are relevant Schouten identities in dimensions d ď 7. Therefore, the analysis of quartic order of interactions becomes more involved. We plan to address that problem both in three-dimensional and higherdimensional contexts in the future.
We delegate some more technical discussion to appendices. In Appendix A, we elaborate on the possibility of writing parity-even vertices as ratios which are by themselves meaningless expressions, but can be defined and motivated only in three dimensions due to their equivalence to vertices of [1] via Schouten identities. We study two-dimensional vertices in Appendix B. There, the restrictions imposed by Schouten identities are much more severe and eventually allow for only vertices of the type ps, s, 1q and ps, s, 0q.

A Vertices as Ratios
In generic dimension, the ps 1 , s 2 , s 3 q cubic interactions can be written in the basis of cubic vertices of the form 10 V s 1 ,s 2 ,s 3 " where g n are undetermined constants. The n´th term in the sum has s 1`s2`s3´2 n derivatives. Therefore, the term with the minimal number of derivatives is n " minps 1 , s 2 , s 3 q while n " 0 contains the maximal number of derivatives. These two bounds are commonly referred to as the lower and upper Metsaev bound respectively. 10 One easily checks that si corresponds to the spins of the field φi by rewriting each term in Vs 1 ,s 2 ,s 3 as where c α,β,γ are the trinomial coefficients. Counting the powers of say a1 then gives s1´n loomoon y s 1´n 1`α lo omo on py 1 z 1 q α "y α 1`β lo omo on For a given dimension, one may be able to construct vertices which violate these bounds due to the presence of Schouten identities. As an example, one can construct vertices corresponding to minimal ps, s, 2q coupling to gravity in three dimensions. These vertices only involve two derivatives and therefore violate the lower Metsaev bound for s ą 2. One may try to express these vertices in the form of (A.3) 11 .
We start from which by itself is of course a nonsensical expression. However, as we will see shortly, one can make sense of it only in three dimensions using Schouten identities. Note that this vertex only contains two derivatives and obviously fulfills D i V " 0. By using the definition of G one obtains The last term however vanishes due to the Schouten Identity (3.10a). Therefore, we have constructed a gauge invariant ps, s, 2q TT vertex involving only two derivatives as can be seen as follows 0 " D i V " D i pV min`∆ˆR q " D i V min`Di p∆qˆR`∆ˆD i pRq. (A.5) Since D i pSchouten Identitiesq Ă Schouten Identities, it then follows that D i V min " 0. This procedure can fail in subtle ways. To illustrate this, let us consider one-derivative minimal coupling to Maxwell field which is defined, similarly to (A.4), through a ratio: it follows for example that D 2 p∆qˆR " pG´y 3 z 3 qy 2 y 3ˆ1 y 3 s´2 ÿ k"0ˆs k`2˙p G´y 3 z 3 q k z s´k`2 3 y k 3 , (A.9) 11 Similar attempt has been made for four-dimensional vertices in [38]. and due to the pole in y 3 the first term in the sum is no longer proportional to a Schouten identity after canceling terms common to the denominator and numerator. Note that such a pole would not arise for the case of minimal coupling to gravity (A.4). It is interesting to notice that the same ratio (A.7) defines a minimal coupling to Chern-Simons field given in (5.5): V min " y 3 z s 3 . (A.10) This is due to first identity of (5.3) which allows to replace (6.1) by ∆ " G´y 3 z 3 " 0 , R " and extending the argument given for spin two minimal coupling to this case. As we have seen, the schematic way of writing the vertex as ratios works consistently only for vertices that are otherwise shown to exist in covariant formulation due to Schouten identities. We conclude that this rewriting is just a curiosity and does not provide with any new insights.

B d=2
In two dimensions the Schouten identities can be used to eliminate the d'Alambertian term in the Fronsdal action for s ě 2 in favour of trace and divergence terms, which renders the free theory to be trivial for TT fields. For example, the spin two Fronsdal equation itself is proportional to Schouten identity. This is a 'linearization' of the statement that Einstein-Hilbert action is topological and there are no Einstein equations of motion for the metric in 2d. For s ą 2, the massless equation (2.2a) is a consequence of the two other equations (2.2b) and (2.2c). One may study massless HS fields in fully reducible Maxwell-like formulation in d " 2. There, each even rank field carries a single scalar mode. HS theories in 2d are making use of BF-type actions [76]. Nevertheless, if one insists on Fronsdal formulation and tries to derive TT cubic couplings, following observations are in order. Some of the Schouten identities that can be derived in this case are: y 1 y 2 y 3 " 0 , y 2 i y j " 0 , y i y j z j " 0 , G 2 " 0 , (B.1) z i pG´y i˘1 z i˘1 q " 0 , z 1 z 2 z 3 " 0 .

(B.2)
These identities imply that any term with more than two derivatives is TT trivial and there are no candidate TT expressions for s 1 ě s 2 ě s 3 ě 2. The only parity-even TT vertices that can be written down, necessarily have scalar or Maxwell fields involved and are given by ps, s, 0q vertex V ps,s,0q " y 1 y 2 z s´1 3 , ps, s, 1q vertex V ps,s,1q " y 3 z s 3 (s ě 2, s " 0) and usual Yang-Mills p1, 1, 1q vertex V p1,1,1q " G. These observations may be useful in the attempts to construct HS gravity theories in two dimensions.