Feynman rules for higher-spin gauge fields on AdS$_{d+1}$

We determine the Feynman rules for the minimal type A higher-spin gauge theory on AdS$_{d+1}$ at cubic order. In particular, we establish the quantum action at cubic order in de Donder gauge, including ghosts. We also give the full de Donder gauge propagators of higher-spin gauge fields and their ghosts. This provides all ingredients needed to quantise the theory at cubic order.


Introduction
The Fronsdal program has been formulated to define consistent non-linear field theories which include interacting higher-spin gauge fields at the classical level [1]. Its long standing motivation has been a deeper understanding of the symmetries behind quantum gravity. Numerous attempts have been devoted to find solutions to this problem from various perspectives, which include the Noether procedure (see  and references therein, for an incomplete list of relevant works), the frame-like formalism [30][31][32][33][34][35][36], and key attempts [37][38][39][40][41][42] to obtain directly a fully non-perturbative formulation of a higher-spin theory.
However, in spite of the above remarkable efforts, all attempts are being confronted with one and the same conceptual subtlety, which is intimately related to the definition of a nonlocal extension of the classical field theoretic deformation problem that lies at the basis of Einstein General Relativity and QFT.
So far, it has been possible to make sense of pseudo-local higher-spin Lagrangians and equations of motion up to the cubic order, where cubic couplings have been recently fixed completely both using holography [43][44][45][46][47][48][49][50] and Noether procedure [27,51]. However, beyond the cubic order a proper extension of the functional class of local Lagrangian functionals and equations of motion is currently lacking. This goes hand in hand with the proliferation of infinitely many explicitly non-local off-shell solutions to the Noether procedure, 1 which lead to one and the same observable defined by AdS/CFT correspondence. It was further argued in [51] that no proper extension of the functional space of non-localities is possible in a properly defined generalised field theoretic context and that one may have to resort to String Theory, i.e. beyond the realm of field theory, to achieve a proper definition of higher-spin theories.
On the other hand any CFT defines, up to off-shell ambiguities, a formal bulk field theory. From this perspective AdS/CFT acquires a tautological meaning and can be considered as some kind of general non-local map/transform which can be inverted to fix bulk Lagrangian couplings in a process that has been referred to as holographic reconstruction [45,[48][49][50][57][58][59]. Exactly in the same way as it is possible to enlarge the functional domain to define an integral transform, the key question is about a clever choice of regularity conditions which ensure a proper independent definition of both the boundary and bulk sides of the duality.
For these reasons it is important to push beyond tree-level, and investigate quantum properties of higher-spin gauge theories independently on both the bulk and boundary sides to test the degree of non-localities. With this motivation in mind, the aim of this work is to revisit the ambient space formalism and formulate a consistent scheme to quantise higherspin gauge theories on AdS by writing down their quantum action in a fixed gauge. Together with recently developed tools [60] to evaluate loop diagrams with external legs on AdS, 2 this constitutes a key step towards quantum tests of the duality, beyond the one-loop vacuum energy checks [64][65][66][67][68][69][70][71][72][73][74][75] which only probe the free theory. In particular, in this work we write down the full cubic quantum action for the type-A theory in the de-Donder gauge and invert the corresponding kinetic terms for both ghosts and physical fields in the same gauge. We also outline a systematic procedure to formally reconstruct the full quantum action at any order in weak fields given the physical vertex for traceless and transverse fields. In the next section we provide a short summary of the main results.

Summary of results
In this work we determine the gauge fixed path integral of the minimal type A higher-spin gauge theory on AdS d+1 up to cubic order fluctuations, together with associated propagators in the same gauge. The path integral reads The free minimal type A Fronsdal action is given by (1.4) where S (2) [ϕ s ] is the action (2.39) describing a free totally symmetric spin-s Fronsdal gauge field ϕ s [76]. The field ϕ 0 is a parity even scalar of fixed mass m 2 0 = −2 (d − 2) /R 2 . This is the minimal spectrum compatible with global higher-spin symmetry [38,77,78].
In §3.1 we derive the cubic order action in de Donder gauge, which reads where [27,48] (using the ambient space formalism reviewed in §2.1): are the cubic couplings for traceless and transverse fields, with coupling constants [27,48] . (1.7) Their de Donder gauge off-shell completion is given in the ambient formalism by which are fixed by requiring gauge invariance in the de-Donder gauge. The free ghost action can be fixed by the linear gauge transformations, and is given by (see §3.2 and, for notation, §2) where c s−1 ,c s−1 are the ghosts associated to the Fronsdal field ϕ s . Likewise, the cubic action is fixed by the first order deformation of the linearised gauge transformations, and reads and which come from the first order deformation (3.26) of the linearsied gauge transformation induced by the de Donder gauge cubic couplings (1.5).
Having gauge fixed the action, in §4.2 we also determine the complete form of the bulk-to bulk propagators for Fronsdal fields and ghosts in the de-Donder gauge, completing the results of [44] to include the gauge terms required for an off-shell source.
The bulk-to-bulk propagator for a spin-s Fronsdal field in de Donder gauge is given by where we decompose in traces. The traceless components are given in spectral form by (1.14d) in terms of harmonic functions Ω (see §B), whose coefficients read The function c l−2 (ν) is given by (B.4). The ghost bulk-to-bulk propagator is similarly given by: (1.17) We also determine the ghost bulk-to-boundary propagators in §4.1.2, which in the ambient space formalism read and normalisation (1. 19) These are to accompany the bulk-to-boundary propagators for the associated spin-s gauge fields, which were determined by Mikhailov in [79]. It is interesting to notice the factor of X 2 appearing in the bulk-to-boundary propagator for the ghosts. Although this factor does not affect the intrinsic projection, this is the correct ambient representation compatible with the fact that in our ambient space conventions ghost do not uplift directly as ambient space harmonic functions. This is due to the leftover mass term present in the ambient equations (2.26). At the practical level one can reabsorb the additional factor of X 2 of (1.18) in the vertices redefining the fields in terms of harmonic representatives allowing a more uniform notation. However, since this issue is not important for the discussion of the present paper we do not discuss it further here.

Notation, conventions and ambient space
In this work we consider higher-spin gauge theories in Euclidean anti-de Sitter (AdS d+1 ) space, where the boundary dimension d is general.
Throughout we employ an operator formalism to manage the tensor indices (for a review see e.g. [80], whose conventions we adopt throughout), where fields are represented by generating functions. For example, a totally symmetric rank-s bulk field ϕ µ 1 ...µs is represented as where we introduced the constant (d + 1)-dimensional auxiliary vector u µ . In packaging totally symmetric tensor in generating functions as above, the action of the covariant derivatives is defined as a differential operator in both x µ and u µ : where u a = e a µ (x) u µ with viel-bein e a µ (x), ω ab µ is the spin connection and L a b the Lorentz tensors which are given by In the following we shall work with contracted auxiliary variables u µ = e µ a (x) u a and the associated derivative ∂ u µ = e a µ (x) ∂ u a . As a consequence of the vielbein postulate: The operator formalism is useful since it allows to translate tensor operations in terms of an operator calculus, which simplifies manipulations. For instance, the contraction: (2.5) and the operations: divergence, symmetrised gradient, box, symmetrised metric, trace and spin are represented by the following operators: divergence: ∇ · ∂ u , sym. gradient: u · ∇, box: , In §A.1 we give the operator algebra.
In this formalism, the usual Fierz-Pauli conditions for a symmetric bosonic spin-s field of mass m 2 s R 2 = ∆ (∆ − d) − s on AdS d+1 are given by: For ∆ = s + d − 2, ϕ s is a gauge field and the system (2.7) is invariant under the gauge transformation δ with symmetric traceless rank-(s − 1) gauge parameter which is on-shell: . Fields which are symmetric and also traceless may furthermore be encoded in generating functions (2.1) of a null auxiliary vector w 2 = 0. In the operator calculus (see e.g. [24,81] and references therein) one replaces the partial derivative ∂ w with the Thomas derivative [82]: which preserves w 2 = 0. In this case the operator calculus simplifies to just four operators: In §A.2 we give the corresponding operator algebra.
On the conformal boundary of AdS d+1 , operators of non-trivial spin can likewise be encoded in generating function notation. A totally symmetric spin-s operator O i 1 ...is at the boundary point y i , i = 1, ..., d can be packaged as with the null auxiliary vector z 2 = 0. In this case the Thomas derivative is: 3 (2.14)
The totally symmetric spin s field ϕ µ 1 ...µs of mass m 2 subject to the following constraints [85]: • Tangentiality to surfaces of constant ρ = √ −X 2 : • The homogeneity condition: where we are free to choose either µ = ∆ or µ = d − ∆. In this work we take µ = ∆.
These constraints make sure that the ambient uplift of fields that live on the AdS manifold is well-defined and 1:1. Like in the previous section, in the above we introduced a generating function to represent the ambient field ϕ A 1 ...As : with constant ambient auxiliary vector U A . The above discussion extends to differential operators. In the operator formalism, the ambient space representative ∇ A of the Levi-Civita connection ∇ µ on AdS d+1 is given by [86][87][88]: In this framework, the intrinsic Fierz-Pauli system (2.7) is described by supplemented with the tangentiality and homogeneity conditions (2.17) and (2.18). For a spin s gauge field ∆ = s + d − 2, the gauge transformation (2.8) reads where the ambient representative of the gauge parameter is subject to the tangentiality and homogeneity conditions and the on-shell constraints (2.10) are represented as: It is straightforward to verify that (2.24) under the constraints (2.25) and (2.26) leaves the Fierz system (2.23) invariant. Traceless fields living on the AdS manifold are represented by traceless ambient representatives with respect to the ambient metric η AB , which themselves can be encoded in generating functions (2.19) with null ambient auxiliary vector W 2 = 0. The Thomas derivative (2.11) reads∂ It is also sometimes useful to impose the constraints: which take care of the tangentiality condition (2.17). Preserving the above constraints in the operator calculus (which is given in §A.3 and §A.4) requires the following modifications of ∂ U and the Thomas derivative (2.27): with projector (2.21).

The AdS boundary
The ambient formalism can also be extended to the AdS boundary [84,85,[89][90][91][92][93]. As the boundary is approached, the hyperboloid (2.15) asymptotes to the light-cone. This limit does not yield a well-defined boundary metric, but one can obtain a finite limit by considering a projective cone of light-rays: Because X 2 is fixed, these null co-ordinates satisfy: and are identified with the AdS boundary. For Euclidean AdS in Poincaré co-ordinates x µ = z, y i , we have: and the boundary points are parameterised by the Poincaré section: A symmetric spin-s boundary field f i 1 ...is (y) of scaling dimension ∆ is assigned an ambient representative f A 1 ...As (P ), which is traceless with respect to the ambient metric 4 and scales as Like for the ambient description of bulk fields detailed above, we require that f A 1 ...As is tangent to the light-cone: However, because P 2 = 0, there is an additional redundancy which, combined with (2.36), eliminates the extra two degrees of freedom per index of f A 1 ...As . The operator formalism also extends to ambient boundary fields, where we have: with the null ambient auxiliary vector Z 2 = 0 imposing the traceless condition (2.34) and it is useful to introduce the constraint P · Z = 0 which implements tangentiality to the light-cone (2.36).

Functionals in de Donder gauge
In this paper we are interested in providing a convenient formalism to deal with AdS Feynman rules for higher-spin gauge fields in the de-Donder gauge, including ghosts. It is thus a key step to describe in detail the corresponding tensor calculus. An off-shell Langrangian for a spin-s gauge field ϕ s freely propagating on AdS d+1 is given by the Fronsdal action [76], which in the operator formalism reads and gauge transformation: with symmetric and traceless rank s − 1 gauge parameter: The equation of motion derived from the free action (2.39) is given by In this work we consider Fronsdal fields in the de Donder gauge: where the equation of motion (2.44) takes the form and can be inverted off-shell. The residual gauge freedom is given by (2.42) with on-shell gauge parameters: Its inverse on doubly traceless fields can be easily computed, and is given by . Any further gauge fixing would thus be an on-shell gauge fixing. For example, the remaining freedom (2.47) can be used to eliminate the trace of the de Donder field (2.45) on-shell, such that it becomes transverse and traceless (T T ): This recovers the Fierz Pauli system (2.7). It is sometimes useful to express the double-traceless Fronsdal field in terms of its irreducible components In the de Donder gauge, the two traceless fields ϕ ′ s andφ s completely decouple, with only the de Donder gauge condition (2.45) relating them. The equation of motion (2.46) decomposes as: de Donder functionals in ambient space The double traceless of the Fronsdal field also extends to its ambient representative: where the ambient representative of the gauge parameter (2.47) is subject to the on-shell constraints Both components are subject to the same homogeneity and tangentiality conditions (2.18) and (2.17). In de Donder gauge, they are governed by the ambient equations of motions 3 Off-shell cubic couplings in de Donder gauge

Fronsdal field cubic couplings
The cubic action for traceless and transverse fields (2.48) in the type A higher-spin gauge theory on AdS d+1 is given in the ambient space formalism by [27,48] and with coupling constant [27,48] g s 1 ,s 2 , .
where N is the only free parameter and is related via holography to the number of CFT degrees of freedom. The off-shell completion of the traceless and transverse cubic action (3.1a) can be determined using the Noether procedure, which at the cubic order requires: which we may solve in de Donder gauge for S (3) using the traceless and transverse action (3.1a) as a starting point.
Since the de Donder gauge condition (2.45) allows to replace gradients with traces, S (3) can be decomposed in traces of the constituent fields: for some functions f s i ,s j and f s 1 ,s 2 ,s 3 of the operators (3.2), which we determine in the following.
For all three constituent fields in de Donder gauge, variation of the TT vertex (3.1b) under a linearised spin-s 1 gauge transformation is given by where for convenience we introduced the notation: for some field f (X, U ) in ambient space. In (A.3) we give some useful identities. The first line of the variation (3.8) is the standard off-shell transformation for traceless and transverse fields [25,27], while the remaining terms denote the corrections to the latter in de Donder gauge. The approach we take to determine the off-shell cubic couplings (3.6b) is to begin with traceless and transverse fields (2.48) and uplift them to de Donder gauge one-by-one. We express this schematically as follows: where the notation −→ ϕs i signifies solving (3.5) at each step for the Fronsdal field ϕ s i in de Donder gauge. In taking this approach to solve at a given order n in traces for V (n) s 1 ,s 2 ,s 3 , the corrections V (m>n) s 1 ,s 2 ,s 3 that are higher order in the traces of the fields do not contribute since only n fields are uplifted to de Donder gauge and the rest are kept traceless and transverse.
We thus first solve for V s 1 ,s 2 ,s 3 where ϕ s 1 is in de Donder gauge with ϕ s 2 and ϕ s 3 traceless and transverse. In this case, the first line of the transformation (3.8) of the TT vertex is vanishing on the free equations of motion of ϕ s 2 and ϕ s 3 , while the final two lines are vanishing owing to the traceless and transverse conditions (2.48). The term on the second line is nonvanishing and must thus be cancelled by the variation of V (1) s 1 ,s 2 ,s 3 . Since the term to be cancelled is proportional to the divergence of the gauge parameter, the appropriate counter-term is simply given by: By cyclising, we thus have: We can now proceed to 2 and solve for V s 1 ,s 2 ,s 3 by compensating terms coming from the variation Like in the previous step, since each term is proportional to the divergence of the gauge parameter we can straightforwardly write down the counter-term which gives Proceeding to the final step 3 , we determine V s 1 ,s 2 ,s 3 by cancelling terms coming from the variation: This requires: which gives the complete cubic coupling (3.6b) in de Donder gauge. The variation is given by: 18) which is proportional to the equation of motion (2.53) for Fronsdal fields in de Donder gauge and thus vanishes on-shell, as required. The operatorsT 12 andT 13 finally read: and Establishing the results of this section made extensive use of the following identity that holds and the identities (A.9).

Ghost cubic couplings
Using the standard Faddeev-Popov procedure, ghost fields are introduced upon exponentiating the determinant in the gauge fixed path integral: In the above we used ϕ to collectively denote the Fronsdal fields in the spectrum of the type A higher-spin gauge theory, subject to the de Donder condition (2.45). Thec, c collectively denote the corresponding ghost fields, which are traceless owing to the tracelessness of Fronsdal gauge parameters (2.42). 7 Where we introduced the operator Q = − 1 The ghost action (3.23) is entirely specified by the non-linear gauge transformations δ ξ of the Fronsdal fields. In particular, expanding (3.23) up to cubic order we see that the cubic vertices of the ghost action are determined by the first order deformation δ Combined with (3.18), the cubic consistency condition (3.4) requires: whereT 13 andT 12 were defined in the linearised variation (3.18) of S (3) . For the ghost cubic action (3.24c), we thus have where we used the identity: The O(u 2 ) terms do not contribute owing to the tracelessness of the ghost fields. can be constructed from the bulk-to-boundary propagator:

Propagators
where: Since the equation (4.1) for the bulk-to-boundary propagator is source-free, we can go on-shell and choose the traceless and tranverse gauge (2.48) using the on-shell gauge parameters (2.47). The equation of motion for the propagator becomes: with boundary condition (4.5b) unchanged. It is most straightforward to solve for the bulkto-boundary propagator in ambient space, in which the ambient representative of the bulk-toboundary propagator satisfies ∂ 2 X K s (X, U ; P, Z) = 0, (4.7a) (∂ U · ∂ X ) K s (X, U ; P, Z) = 0, (4.7b) Together with the tangentiality and homogeneity conditions: (X · ∂ U ) K s (X, U ; P, Z) = (X · ∂ X + s + d − 2) K s (X, U ; P, Z) = 0, (4.8a) (Z · ∂ P ) K s (X, U ; P, Z) = (P · ∂ P + s + d − 2) K s (X, U ; P, Z) = 0, (4.8b) the solution is fixed uniquely up to an overall coefficient [79]: The coefficient is fixed by equation (4.5b) to be: (4.10) 8 For concreteness, here we used co-ordinates x µ = y i , r for AdS d+1 : with R the AdS radius. The boundary of AdS is located at r = ∞, with boundary directions y i , i = 1, ..., d.
In ambient space it is straightforward to extend the above result to the bulk-to-boundary propagator K ∆,s of a totally symmetric spin-s field of generic mass m 2 R 2 = ∆ (∆ − d) − s, which has the same ambient equation of motion (4.7) but with the homogeneity degree: The result is simply [79,96]: (4.12b) which coincides with (4.9) for ∆ = s + d − 2, as required.
To summarise, the bulk-to-boundary propagator for a ghost field associated to a spin s gauge field is given in ambient space by where C d+s−1,s−1 is defined in (4.12b).

Bulk-to-bulk propagators of Fronsdal fields in de-Donder gauge
The solution to the Euler-Lagrange equations for the Fronsdal action (2.39) with some source J s , 9 can be expressed in terms of the bulk-to-bulk propagator Π s , which satisfies the simpler equation where Λ s,s−1 is a pure gauge term, subject to the constraints The pure gauge term Λ s,s−1 is is often disregarded when the source J s is conserved. 10 In the context of Witten diagrams, this is the case when the constituent fields of J s are external and thus on-shell. See [44,45,49,50,63,97,98] in the context of the four-point tree-level exchange of a spin-s gauge field. However when the source is off-shell, such as in loop diagrams, this no longer holds and the explicit form of Λ s,s−1 is required. In the following we complete the results of [44,63,97,98] and determine the full spin-s bulk-to-bulk propagator (4.20) in de Donder gauge, including the form of the pure gauge terms Λ s,s−1 . To this end, it is useful to express the double-traceless Fronsdal field in terms of its irreducible components (2.49), for which it is useful to employ the notation where we have introduced a traceless auxiliary variable w with the property w 2 = 0, which ensures the tracelessness of each component. See §2. In this notation, the Fronsdal bulk-to-bulk propagator is thus a 2 × 2 matrix which involves off-diagonal mixing terms between the two off-shell irreducible components of a doubly traceless Fronsdal field. In the de Donder gauge, the gauge fixed equation of motion for the bulk-to-bulk propagator is 11 (4.24) and propagator must also satisfy the de Donder condition: To solve for the propagator it is most straightforward to use ambient space with constrained auxiliary variables (2.28). In terms of the latter, and furthermore focusing on the irreducible components (4.23), the de Donder condition (4.25) reads: 26) where the zeros on the second line are owing to the tracelessness of the de-Donder operator.

Bulk-to-bulk propagators of ghost fields
The equation for the bulk-to-bulk propagator of a ghost field c s−1 associated to a spin-s Fronsdal field in de Donder gauge can be derived from the Euler-Lagrangian equations of the free ghost action (3.24b). The resulting propagator equation is: where we recall that m 2 ξ R 2 = (s − 1) (s + d − 2). The ghost fields and thus their bulk-to-bulk propagators are traceless, so the harmonic function decomposition of the latter takes the form: (4.36)

Beyond cubic order
We conclude presenting technical trick which, given the gauge-fixed de Donder gauge vertices of Fronsdal fields, allows to obtain the corresponding ghost vertices directly in the ambient space formalism. This trick is shown to work at any order in the weak fields once the corresponding de-Donder completion of the vertices is obtained.
The key observation is the identity: from which, keeping track of normalisations, one obtains the following ghost vertex in terms of the variation of the physical vertex: s AdS d+1 e ∂u 1 ·∂u 2 (u 1 · ∇ 1cs−1 ) u 1 δ δξ s−1 T (ϕ, ξ) c s−1 ϕ u 2 .

(5.2)
The observation is that the above ghost action is exactly proportional to the gauge variation of the de-Donder vertex up to a simple substitution, which can be identified as: where we have used that the degree of homogeneity of U 3 · ∇ 3 c 3 is the same as that of ϕ together with the fact that both U 3 · ∇ 3 c 3 and X 2 3 F 3 ϕ 3 are tangent tensors. In particular, we arrive to the following list of substitutions: which trivially allow to recover the ghost vertex from (3.19) and (3.20). The latter substitutions generalise to all orders in the Noether procedure. Indeed the n-th order deformation of the gauge transformation has also the same general structure as above: One can then obtain the ghost vertex at order n simply from the above right-hand side, with the substitutions (5.4).

A.4 Traceless ambient fields
Here we give the operator algebra for the null auxiliary ambient vector W 2 = 0 subject to the tangentiality constraint X · W = 0. We furthermore restrict to the hyperboloid X 2 + 1 = 0.