On Interacting Higher Spin Bosonic Gauge Fields in BRST-antifield Formalism

We examine interacting bosonic higher spin gauge fields in the BRST-antifield formalism. Assuming that an interacting action $S$ is a deformation of the free action with a deformation parameter $g$, we solve the master equation $(S,S)=0$ from the lower orders in $g$. It is shown that choosing a certain cubic interaction as the first order deformation, we can solve the master equation and obtain an action containing all orders in $g$. The anti-ghost number of the obtained action is less than or equal to two. Furthermore we show that the obtained action is lifted to that of interacting bosonic higher spin gauge fields on anti-de Sitter spaces.


Introduction
Higher spin gauge theories have been studied since 1930s from various interests. For example, they are expected to reveal characteristic aspects of string theory in the high energy limit. String theory may be regarded as a spontaneous symmetry breaking phase of the higher spin gauge theories [1]. Free higher spin gauge theories are well understood by now. There are obstacles to construct consistent interactions. One of them is the no-go theorem [2] * . To avoid it, the number of derivatives contained in interaction vertices should be restricted so that higher spin gauge fields are not included in the asymptotic states. For cubic vertices, the allowed number of derivatives are clarified by using a light-cone formulation in [4] [5]. For bosonic gauge fields, vertices are constructed explicitly by using Noether's procedure in [6] [7]. In constructing vertices, a generalized curvature tensor [8] is frequently used as a building block. It is gauge-invariant and defined as s-curl of a totally symmetric spin-s bosonic gauge field. It is not easy task to construct a gauge-invariant generalized curvature tensor on a general backgrounds including anti-de Sitter (AdS) spaces. In contrast we will employ the Fronsdal tensor [9] as a building block, and look for vertices on AdS spaces. It is also difficult to construct full interactions including not only a cubic vertex but also higher order vertices. We usually construct an interacting theory as a small deformation of the free theory including a few lower orders in the deformation parameter g. It may not be consistent to construct vertices beyond the cubic order. Even if we could construct interactions to any order, the full interacting action may not be written in a closed form.
In this paper, we will construct actions of interacting bosonic higher spin gauge fields on D-dimensional spacetimes in the BRST-antifield formalism. The BRST-antifield formalism is known to be very powerful in constructing interactions systematically [10]. Employing this cohomological method, interaction terms are constructed systematically in [11]. In the present paper, at first, we will construct actions on a flat spacetime using this method, and then lift them to those on AdS spaces. For this we will use the Fronsdal tensor as a building block. To avoid the no-go theorem, spins of gauge fields are restricted appropriately. We will comment on this point in the last section. In section 4, two gauge fields of spin-s and spin-2s are examined, and three gauge fields of spin-s 1 , spin-s 2 and spin-(s 1 + s 2 ) are examined in section 5. In either cases, we construct a full action S as a deformation of the free action S 0 , S = S 0 + gS 1 + g 2 S 2 + · · · . It is shown that choosing a certain cubic vertex as S 1 , the master equation (S, S) = 0 can be solved from the lower order in g. It is worth noting that the obtained action S contains all orders in g and is written in a closed form by using a geometric series. The action can be rearranged as S = S 0 + S 1 + S 2 where the anti-ghost * In [3], the properties of bosonic and fermionic particles with spin j ≥ 1 are examined in the field theory framework.
number of S a is a. We extract the gauge symmetry of the interacting action S 0 from the BRST symmetry δ g B X = (X, S). Furthermore, we will write down actions of interacting bosonic higher spin gauge fields on AdS spaces from those on a flat spacetime.
The organization of the present paper is as follows. In the next section, clarifying our notations we present the free action in the BRST-antifield formalism. The BRST deformation scheme is explained in section 3. In section 4, two interacting gauge fields with spin-s and spin-2s are examined. The master equation is solved from the lower orders in g and the full action containing all orders in g is derived. In section 5, the full action for three interacting gauge fields with spin-s 1 , spin-s 2 and spin-(s 1 + s 2 ) is derived (a derivation of low order deformations is given in appendix B). In section 6, we will write down actions of higher spin bosonic gauge fields on AdS spaces. The last section is devoted to a summary and discussions. In appendix A, we explain how a Γ-exact term results in a BRST-exact term.

Free higher spin bosonic gauge theory
We will explain our notations used in this paper and present the free action in the BRSTantifield formalism.

Free action in the BRST-antifield formalism
Corresponding to the gauge parameter ξ µ 1 ···µ s−1 , we introduce a rank-(s − 1) Grassmannodd ghost field c µ 1 ···µ s−1 with the same algebraic symmetry. The c µ 1 ···µ s−1 must be traceless The gauge field φ and the ghost field c are collectively called "fields" and denoted as Φ A .
Z pgh(Z) agh(Z) gh(Z) We further introduce "antifields" Φ * A = {φ * µ 1 ···µs , c * µ 1 ···µ s−1 } which have the same algebraic symmetries but opposite Grassmann parity. Two gradings are introduced. One is the pure ghost number pgh, and the other is the antighost number agh. The ghost number gh is defined as gh ≡ pgh − agh . The grading properties are summarized in the Table.1. The antibracket for two functionals, (2.12) The action S dWF in (2.6) can be extended to S 0 [Φ, Φ * ] such that the BRST transformation of a functional X(Φ A , Φ * A ) is expressed as Note that δ B acts from the right. The nilpotency δ 2 B = 0 requires the master equation (S 0 , S 0 ) = 0 . In the present case, the free action is found to be which leads to (2.10), (2.11) and Note that the second term in the right hand side of (2.16) is required for the nilpotency δ 2 B c * = 0. It is straightforward to see that this term leaves the action (2.14) unchanged.

BRST deformation
The BRST deformation scheme is very useful in constructing interactions systematically [10]. We will explain relevant aspects of it in this section. Suppose that S is a deformation of S 0 expanded in a deformation parameter g S = S 0 + gS 1 + g 2 S 2 + · · · . The equation for n = 0, (S 0 , S 0 ) = 0, is satisfied by definition. It is worth noting that S n is determined by all of the lower order terms, S k (k < n). To obtain higher order terms, we must start examining S 1 first of all.
The S 1 is determined by the equation (3.2) for n = 1 , which can absorb (X, S 0 ) by choosing u A and v A appropriately.
In this paper, we will choose a cubic interaction as S 1 . In this case, S 1 can be expanded in the antighost number as where agh(a i ) = i. Let us explain the reason why a n (n > 2) are excluded. Since the action S has ghost number 0, gh(a i ) = 0, so that pgh(a i ) = agh(a i ) = i. As pgh(a 3 ) = 3, a 3 must be composed of three c's. But such an a 3 has gh(a 3 ) = 0. This is because we exclude a 3 in the expansion. For a n (n > 3), pgh(a n ) = n. It is impossible to construct such a term as a cubic interaction. Similarly we expand δ B with respect to the agh as where agh(∆) = −1 and agh(Γ) = 0. The ∆ and Γ are nilpotent and anticommute each other. These differentials and antifields act on fields as summarized in the Table.2. It is easy ‡ Let a be a D-form defined by S 1 = a, and then this is equivalent to δ B a + db = 0 . Since a BRST exact part of a corresponds to a trivial field redefinition, we consider H 0 (δ B |d): the cohomology of the BRST differential δ B modulo d at gh = 0.
Since agh(a 0 ) = 0, a 0 does not contain any antifields Φ * A . It implies (3.10). First we consider (3.7). Since a 2 has agh = pgh = 2, it is composed of two ghosts c's and one anti-ghost c * . For a non-trivial a 1 , ∆a 2 has to be Γ-exact as seen from (3.8). It implies that ∂c must be included in a 2 such as d D x c * µ 1 ···µpρ 1 ···ρr c ν 1 ···νq This means that a 2 is Γ-exact. As is explained in appendix A, such a Γ-exact a 2 leads to a BRST-trivial S 1 . So we may set We construct a 1 satisfying Γa 1 = 0. Since a 1 has agh = pgh = 1, it is composed of a ghost c, an antifield φ * and a gauge field φ. For Γa 1 = 0, φ should appear as a gauge invariant form such as § G(φ), since ΓG(φ) = 0 which follows from δ B F = 0. In addition, as a 1 , which is not Γ-exact. This is the a 1 from which all interaction terms are shown to be constructed successively. In this paper, we concern a special case with r = 0. Even for the case with r = 0, we have found that the S n can be constructed successively. We hope that we will report this result in another place [12]. § One may use F instead of G, because the difference between a 1 with G and a 1 with F is absorbed by a field-redefinition. In fact, the difference results in a term proportional to ∂ · c which is just a Γ-exact term.
In the followings, we will construct a BRST-invariant action S containing all orders in g. In the next section, we consider two interacting gauge fields with spin-s and spin-2s. In section 5, we generalize it further to the case with three interacting gauge fields with spin-s 1 , spin-s 2 and spin-(s 1 + s 2 ).
4 Interaction of two gauge fields with spin-s and spin-

Three-point interaction
As explained in section 3, we choose the Γ-nontrivial a 1 as which is (3.12) with r = 0. We will show that all higher order terms can be constructed from this a 1 . Here we have introduced∂c = ∂ (µ 1 c µ 2 ···µs)ν 1 ···νs . We note ∂c in (4.2) is a symmetric matrix but∂c is not. In fact, we note that instead of (4.3), so that the obtained action S should be the same subject to a field redefinition. This is because the difference between (4.3) and (4.4) is a Γ-exact term, It is obvious that (4.3) satisfies Γa 1 = 0 as ΓG(φ) = 0 which follows from the gauge invariance of F (φ). To solve (3.9) for a 0 , we derive It implies that we may determine a 0 as As a result, we have obtained S 1 in (3.5) composed of (3.11), (4.3) and (4.6).

Four-point interaction
We will proceed to derive S 2 . The S 2 is determined by the master equation (3.2) for n = 2 It implies that S 2 is composed of four fields because S 1 is composed of three fields. Expanding S 2 with respect to agh as where agh(b i ) = i, we find that (4.7) reduces to Γb 2 =0 , (4.9) We have not included b n (n > 2) in the expansion (4.8) simply because we can solve (4.7) without them. First we will solve (4.10) for b 2 and b 1 . One derives using a 1 in (4.3) In the second equality, we have used ∂c =∂c + (∂c) T . We find that the first term in the last line is Γ-exact while the second term is ∆-exact where we have used (2.8). As a result, we obtain It is obvious to see that b 2 above satisfies (4.9). Next we will solve (4.11) for b 0 . Noting that where (2.8) is used in the last equality, we find that As a result, we found that S 2 in (4.8) is composed of (4.15), (4.16) and (4.19).

Five-point interaction
In order to guess the form of S n , let us proceed to derive S 3 . The Expanding S 3 with respect to agh as where agh(c i ) = i, we find that (4.20) reduces to As seen below, we can solve (4.20) without c n (n > 2), so that they are not included in (4.21). First we will solve (4.22) for c 2 . One derives using (4.3) and (4.15) where in the second equality,∂c + (∂c) T = ∂c is used. This implies that Next we solve (4.23) for c 1 . For this purpose we derive Combining the second term in the right hand side of (4.28) with the right hand side of (4.29), we obtain On the other hand, the first term in the right hand side of (4.28) may be rewritten as so that this term and the right hand side of (4.27) make d D xG(∂cφ * ) T φG(∂cG(φ)) . Because (4.32) cancels out the last term in (4.30), we conclude that which implies that Finally, (4.24) is examined to determine c 0 . It is straightforward to see that where in the second equality we have used (2.8) and transposed the integrand, and that Because (4.36) cancels out the last term in the most right hand side of (4.35), we conclude that This implies that As a result, S 3 in (4.21) is found to be composed of (4.26), (4.34) and (4.38).
We shall show that S n (n ≥ 1) above solves the master equation (3.2). Now, suppose that S k (k < n) solves the master equation at the order of g k (k < n). We will solve the master equation (3.2) for α n 2 , α n 1 and α n 0 and show that they coincide with (4.40), (4.41) and (4.42).
The master equation (3.2) at the order of g n is expanded with respect to agh as First we will solve (4.43) for α n 2 . Since we obtain This implies that α n 2 is given as (4.40). Next we will derive α n 1 from (4.44). We observe that where in the second equality we have used a useful relation which follows from (2.8). Furthermore deriving we find that This implies that α n 1 is given as in (4.41). Finally, we derive α n 0 from (4.45). It is straightforward to derive This implies that α n 0 is given as in (4.42). Summarizing the above results, we have shown that S n is surely given in (4.39)-(4.42).

BRST-invariant action of interacting spin-s and spin-2s gauge fields
We will examine the total action S. In the above we have derived S k (k = 1, 2, · · · ). The free action is a sum of S 0 (s) and S 0 (2s) given in (4.1) and (4.2), respectively. By gathering these results together, the total action is given as We find that the action turns to the form expanded in agh as where agh(S i ) = i. This is one of our main results in this paper. The obtained action S contains all orders in g, and BRST-invariant δ g B S = (S, S) = 0. It is not likely that S and S 0 are related each other by a field redefinition as long as we examined.
We have obtained BRST-invariant action of interacting gauge fields by using the BRSTantifield formalism. Here we examine the gauge invariance of the action S 0 . The gauge transformation can be extracted from the BRST transformation δ g B X = (X, S). We find that the gauge transformation with a rank-(s − 1) parameter ξ remains unchanged δφ = ∂ξ , δφ = 0 , (4.59) while the gauge transformation with a rank-(2s − 1) parameter ξ turns to We note that higher order interactions make δφ non-trivial. It is instructive to check the gauge invariance of S 0 in (4.56) under (4.60). The gauge variation of S 0 is The gauge variation of the second term in the right hand side of (4.56) is eliminated by G(∂ξ) = 0. We find that the first term in the right hand side of (4.61) turns to It follows that the right hand side of (4.61) cancels out. Summarizing, we find that the action S 0 in (4.56) is invariant under the gauge transformations (4.59) and (4.60).
In section 6, we will show that the action S obtained in this section can be easily generalized to the action on AdS spaces. In this section, we will slightly generalize the model examined in the previous section. We introduce three gauge fields with spin-s 1 , spin-s 2 and spin-(s 1 + s 2 ). The spin-s I (I = 1, 2) fields are denoted as d(s I )-component column vectors, such as φ µ 1 ···µs I = φ (I) , φ * µ 1 ···µs I = φ * (I) and ∂ (µ 1 c µ 2 ···µs I ) = ∂c (I) , while the spin-(s 1 + s 2 ) fields are as rectangular matrices denoted as φ µ 1 ···µ s 1 +s 2 = φ, φ * µ 1 ···µ s 1 +s 2 = φ * and ∂ (µ 1 c µ 2 ···µ s 1 +s 2 ) = ∂c. In this notation, the free action is written as where the summation over I = 1, 2 is understood. As explained above, S 1 is expanded as in (3.5) with a 2 = 0. In the present case, we choose a 1 = d D x G(φ (1) ) µ 1 ···µs 1 ∂ (µ 1 c µ 2 ···µs 1 )ν 1 ···νs 2 φ * (2)ν 1 ···νs 2 + G(φ (2) ) ν 1 ···νs 2 ∂ (ν 1 c ν 2 ···νs 2 )µ 1 ···µs 1 φ * (1)µ 1 ···µs 1 (5.2) as a 1 . We will show that this a 1 leads to an action of interacting three gauge fields. For notational simplicity, we write (5.2) as where s IJ denotes a 2 × 2 matrix of s 12 = s 21 = 1 and s 11 = s 22 = 0. In this notation, the index of the derivative in∂c is always contracted with one of the indices of the field sitting to its immediate left. In other words,∂c in (5.3) is a d(s I ) × d(s J ) matrix. We can derive S 1 , S 2 and S 3 in a similar way to the one presented in the preceding section. For the present paper to be self-contained, we give a brief derivation of them in appendix B. From the results obtained there, we can guess the form of S n (n ≥ 1) as follows where Φ −1 = 0 is understood because α 1 2 = a 2 = 0. We note s 2 IJ = δ IJ , s 3 IJ = s IJ , and so on. It is easy to see that S n for n = 1, 2, 3 coincide with those obtained in appendix B. We will derive S n supposing that S k (k < n) solve the master equation at the order of g k (k < n).
The master equation (3.2) is expanded at the order of g n as (4.43), (4.44) and (4.45). First, we shall solve (4.43) for α n 2 . It is straightforward to see that This implies that which coincides with (5.5). Next we will derive α n 1 from (4.44). For this purpose, we derive Gathering these results together, we find This implies that α n 1 is given as in (5.6). Finally, we solve (4.45) for α n 0 . It is now easy to derive This implies that α n 0 is given as in (5.7). Summarizing the above results, we have shown that S n is surely given as (5.4)-(5.7).

BRST-invariant action of interacting gauge fields of spin-s 1 , s 2
and s 1 + s 2 We will examine the total action S. In the above we have derived S k (k = 1, 2, · · · ). The free action S 0 is given in (5.1). Gathering these together, we obtain the total action 14) It is straightforward to see that the action turns to the form expanded in agh as where agh(S i ) = i. This action S contains all order in g, and invariant under BRST transformation δ g B S = (S, S) = 0. This is one of our main results in this paper. We shall examine the gauge invariance of the gauge invariant action S 0 . The gauge transformation can be extracted from the BRST transformation δ g B X = (X, S). We find the gauge transformation with a rank-(s I − 1) parameter ξ (I) remains unchanged while the gauge transformation with a rank-(s 1 + s 2 − 1) parameter ξ turns to It is instructive to check the gauge invariance of S 0 under (5.20). The gauge variation of S 0 is ) .

(5.21)
We find that the first term in the right hand side of (5.21) turns to It follows that the right hand side of (5.21) cancels out. As a result, we find that the action In the next section, we will write down an action of interacting three gauge fields on AdS spaces from the action obtained in this section.

Interacting higher spin gauge fields on AdS spaces
We have obtained actions of interacting gauge fields in sections 4 and 5. One of advantages in our approach is that it is easy to generalize the action on a flat spacetime to that on AdS spaces. In this section, we will write down actions of interacting gauge fields on AdS spaces from those obtained in sections 4 and 5.
First of all, we will introduce objects on AdS spaces. The Fronsdal tensor on AdS spaces is given as where = g µν ∇ µ ∇ ν and l denotes the radius of the AdS space. The Fronsdal tensor in (6.1) reduces to (2.1) in the flat limit ¶ l → ∞. The m 2 means Under the gauge transformation δφ µ 1 ···µs = ∇ (µ 1 ξ µ 2 ···µs) , (6.3) ¶ Note that the third term in the right hand side contains a factor 1/2. This is needed for (6.1) to reproduce (2.1) in the flat limit. In fact, the term 1 2 (∇ µ1 ∇ µ2 + ∇ µ2 ∇ µ1 )φ ′ contained in the right hand side of (6.1) becomes ∂ µ1 ∂ µ2 φ ′ in the flat limit. On the other hand, the right hand side of (2.1) contains ∂ µ1 ∂ µ2 φ ′ , which coincides with the flat limit of (6.1).
we obtain This reduces to (2.3) in the flat limit. It follows that the Fronsdal tensor is gauge invariant if ξ ′ = 0. To derive this relation, we have used the fact on AdS spaces (6.5) A useful and important relation is Observe that (6.7) surely reduces to (2.8) in the flat limit. Now we will write down the action of interacting gauge fields on AdS spaces. It is straightforward to obtain the action on AdS spaces from the action obtained in section 4. One obtains S =S 0 + S 1 + S 2 (6.9) , (6.10) On the other hand, from the action obtained in section 5 one obtains 14)

Summary and Discussion
We have constructed actions of interacting bosonic higher spin gauge fields in the BRSTantifield formalism. In section 4, two gauge fields of spin-2s and spin-s are examined, and three gauge fields of spin-s 1 , spin-s 2 and spin-(s 1 + s 2 ) are examined in section 5. In both cases, we constructed an action S as a deformation of the free action S 0 , S = S 0 +gS 1 +g 2 S 2 + · · · . Choosing a cubic interaction as S 1 , we solved the master equation (S, S) = 0 from the lower order in g. We note that our obtained action S contains all orders in g and is BRSTinvariant δ g B S = (S, S) = 0, while the free action S 0 is BRST-invariant δ B S 0 = (S 0 , S 0 ) = 0. Our action is composed of terms with agh ≤ 2, and may be rearranged as S = S 0 + S 1 + S 2 where agh(S i ) = i. From the BRST transformation δ g B X = (X, S), we have extracted the gauge transformation and showed the gauge invariance of S 0 . In section 6, we have written down actions of interacting bosonic higher spin gauge fields on AdS spaces from those obtained in sections 4 and 5.
The obtained action is composed of n + 2 fields and 2n + 2 derivatives at the order of g n . The cubic interaction term contains four derivatives. This is compared with the models composed of the generalized curvature tensor. These models avoid the no-go theorem by restricting the number of derivatives. Our models suffer from the no-go theorem as well. So the spin of the fields should be restricted appropriately [4] as follows: s = 1, 2 for the (s, 2s, s) cubic coupling while (s 1 , s 2 , s 3 ) = (1, 1, 2), (1, 2, 3) and (2, 2, 4) for the (s 1 , s 2 , s 1 + s 2 ) cubic coupling.
Higher spin gauge theories have attracted renewd interests in the study of the AdS/CFT correspondence [13]. Higher spin gauge theories have been conjectured to be dual to simple conformal field theories. ‖ Our models on AdS spaces are much simpler than them and actions are given in this paper. It is interesting to explore the CFT duals to our models.
In this paper, we have examined a special kind of interaction terms, in which each interaction term in S forms an open chain of fields. This enabled us to employ a matrix notation for higher spin fields. It is possible to use this notation even if each interaction term forms a closed chain of fields. In fact, we have found that even in this case we can construct an action of interacting bosonic higher spin gauge theory. We hope to report this result in another place [12]. The most general interaction term may not form a chain of fields but a complete graph (or a simplex), for example a complete graph with four-vertices K 4 (or a tetrahedron) for a four-point interaction. Each vertex represents a field and a edge represents a contraction between two fields. In this case, we must employ a tensor-like notation. This ‖ A bosonic higher spin gauge theory on a three-dimensional AdS space (AdS 3 ) [14] has been conjectured [15] to be dual to the 't Hooft limit of the W N minimal model, while a Vasiliev's higher spin gauge theory on AdS 4 [16] is conjectured [17] to be dual to three-dimensional O(N ) sigma-model. is left for a future investigation.
Another interesting issue to purchase is to include fermionic and bosonic gauge fields with mixed indices. There are some interesting works examining a deformation of the free action with fermions. In [18] a systematic analysis including fermionic gauge fields was performed in the BRST-antifield formalism * * . It is interesting to examine actions of interacting bosonic and fermionic fields including higher orders in g, and to generalize them to those on AdS spaces.
Suppose that a 2 is Γ-exact as is given in (A.1). Substituting this into (3.8), we derive Γa 1 = −∆(Γm 2 ) = Γ(∆m 2 ) , (A.4) so that a 1 turns to the form of (A.2). Similarly, when a 1 takes the form (A.2), substituting it into (3.9) one sees that a 0 takes the form (A.3) . We conclude that a Γ-trivial a 2 leads to a BRST-trivial S 1 . So we may set a 2 = 0. Similarly a Γ-trivial a 1 leads to a BRST-trivial S 1 .
B Derivation of three-, four-and five-point interaction of three gauge fields We will derive S 1 , S 2 and S 3 from (5.3).