Quantum flux operators in higher spin theories

We construct Carrollian higher spin field theories by reducing the bosonic Fronsdal theories in flat spacetime to future null infinity. We extend the Poincar\'e fluxes to quantum flux operators which generate Carrollian diffeomorphism, namely supertranslation and superrotation. These flux operators form a closed symmetry algebra once including a helicity flux operator which follows from higher spin super-duality transformation. The super-duality transformation is an angle-dependent transformation at future null infinity which generalizes the usual electro-magnetic duality transformation. The results agree with the lower spin cases when restricting to $s=0,1,2$.


Introduction
Recently, Carrollian manifolds [1,2] have received much attention due to their relations to null geometries.It has been shown that various physically interesting symmetries could be embedded into the geometric symmetry of Carrollian manifold [3][4][5], including the BMS groups [6][7][8][9][10][11][12], Newman-Unti group, etc..Moreover, the Carrollian diffeomorphism, which preserves the null structure of Carrollian manifolds, is nontrivial [13][14][15] since one can construct corresponding quantum flux operators at future null infinity for lower spin (s = 0, 1, 2) theories.The quantum flux operators are obtained by analyzing the Poincaré flux densities which are radiated to future null infinity.They form a faithful representation of Carrollian diffeomorphism for scalar field theory up to an anomalous term which is the intrinsic central charge of the theory.For massless theories with non-zero helicity, the superrotation calls for super-duality transformation and one should also consider the corresponding helicity flux operators.The results can also be extended to various null hypersurfaces in general dimensions [16].
The structure of the paper is as follows.In section 2, we will introduce the basic ingredients of the Carrollian manifold and review the coordinate systems we adopt in this article.In section 3, we will introduce minimal background on the Fronsdal theory in the flat spacetime.In section 4, we will reduce the bulk HS theories to future null infinity and find the boundary equation of motion as well as the symplectic form.We will construct quantum flux operators and compute the Lie algebra they generate in the following section.The helicity flux operator is discussed in section 6.We will conclude in section 7 and technical details are relegated to several appendices.

Carrollian manifold and coordinate systems
In this work, we will use the Greek alphabet µ, ν, ρ, σ, λ, κ to denote tensor components in Cartesian coordinates.For example, the Minkowski spacetime R 1,3 can be described in Cartesian coordinates x µ = (t, x i ) where µ = 0, 1, 2, 3 denotes the spacetime components and i = 1, 2, 3 labels the spatial directions.We will also use the Greek alphabet α, β, γ, δ to represent components in retarded coordinates.As an illustration, the metric of the Minkowski spacetime in retarded coordinates x α = (u, r, θ, ϕ) is 2) The capital Latin alphabet A, B, • • • will be used to represent the components of tensors on S 2 in spherical coordinates.The future null infinity I + is a three-dimensional Carrollian manifold with a degenerate metric which could be obtained by choosing a cutoff r = R, using a Weyl scaling to remove the conformal factor in the induced metric and taking the limit R → ∞ with the retarded time u fixed.The spherical coordinates θ A = (θ, ϕ) are used to describe the unit sphere whose metric reads explicitly as We will also use the notation Ω = (θ, ϕ) to denote the spherical coordinates in the context.The covariant derivative ∇ A is adapted to the metric γ AB , while ∇ µ adapts to the Minkowski metric in Cartesian frame.The integral measure on I + is abbreviated as where the integral measure on S 2 is dΩ ≡ (2.7) The Levi-Civita tensor on S 2 is denoted as ϵ = 1 2 ϵ AB dθ A ∧ dθ B with ϵ θϕ = −ϵ ϕθ = sin θ, ϵ θθ = ϵ ϕϕ = 0. (2.8) The Dirac delta function on S 2 is (2.9) Besides the metric (2.4), there is also a distinguished null vector which generates the retarded time direction.The Carrollian diffeomorphism is generated by the vector field where f = f (u, Ω) is any smooth function of I + while Y A = Y A (Ω) is time-independent and only smooth vector field on S 2 .The Carrollian diffeomorphism generated by In the following, we may also use stereographic project coordinates on S 2 which are defined by and the metric of (2.13) The volume form reads with the Levi-Civita tensor being The Dirac delta function is defined by .16)In this coordinate system, any rank s symmetric traceless tensor T A(s) can only have two nonvanishing components T z(s) , T z(s) . (2.17) Here we use the short notation to represent a rank s symmetric tensor when it causes no confusion.The element of the permutation group S s is denoted as π in the above equation.The round brackets (• • • ) represent complete symmetrization for the indices inside them.Similarly, the square brackets [• • • ] imply complete antisymmetrization, e.g., We will also use the abbreviation which is a slight abuse of notation.Here the same lower (or upper) indices As are totally symmetrized automatically.One should not be confused with the Einstein summation convention where lower and upper indices are denoted by the same letter.

Metric-like formulation
In this section, we shall review the metric-like formulation of free massless fields of arbitrary spin s.We shall mainly concentrate however only on bosonic fields [67] in flat spacetime, while leaving the fermionic HS fields [68] and HS fields in AdS or dS [69] spacetime for future study [70].As a generalization of the electromagnetism and linearized Einstein gravity, a spin s HS gauge theory (s > 2) is described by a totally symmetric and doubly traceless Fronsdal field f µ(s) where we use a prime to denote the trace of the HS field Therefore, a double prime f ′′ µ(s−4) is the double trace of the HS field A totally symmetric rank s field has4 independent components in general d dimensions.Therefore, the number of independent components of a spin s field is In four dimensions, this number reduces to 2(1 + s 2 ).The spin s field satisfies the Fronsdal equation which is invariant under the linearized gauge transformation where the rank s − 1 tensor ξ µ(s−1) is totally symmetric and traceless The corresponding action is where the Lagrangian density is The action reduces to the Pauli-Fierz action for s = 2 and the Maxwell action for s = 1.The Lagrangian density may be expressed as a compact quadratic form where the rank 2s + 2 tensor L ρµ(s)σν(s) is symmetric in the index sets µ(s) and ν(s) separately.
It is also doubly traceless with respect to these two sets of indices L ′′ρµ(s−4)σν(s) = 0, L ′′ρµ(s)σν(s−4) = 0. (3.12) It may be obtained by taking the symmetric and doubly traceless part of the following rank 2s + 2 tensor with respect to two sets of indices µ(s) and ν(s) separately.
Gauge fixing condition.We may choose the following gauge fixing condition to reduce the Fronsdal equation to The gauge fixing condition (3.14) is always possible.More explicitly, we may start from a general field configuration with G µ(s−1) ̸ = 0 and choose the gauge parameter ξ µ(s−1) such that This is equivalent to the equation whose solution always exists after imposing appropriate initial and boundary conditions.The residue gauge parameter should satisfy the equation which could be used to set the Fronsdal field to be traceless This is always possible since the solution of (3.15) and (3.18) is in terms of plane waves.There is no more constraint on the polarization tensors ε µ(s) and κ µ(s−1) except that κ µ(s−1) is traceless and ε µ(s) is doubly traceless Considering a solution f µ(s) which is not traceless we may always find a tensor κ µ(s−1) such that Therefore, we can always set the HS field to be transverse and traceless.The remaining number of degrees of freedom for the polarization tensor ε µ(s) is Similarly, the remaining number of degrees of freedom for the polarization tensor κ µ(s) is 2s − 1.
We may impose a further condition to reduce the number of degrees of freedom to 2. This is the number of propagating degrees of freedom in four dimensions.When we reduce the theory to future null infinity, the fundamental field F A(s) that encodes the radiation information, has exactly two independent components (seeing the next section).The condition (3.25) in retarded coordinates becomes Such a condition requires which has 2s − 1 components, and will exhaust the degrees of freedom of κ α(s−1) .

Asymptotic equation of motion and symplectic form
Near I + , we may impose the fall-off condition for the HS field in Cartesian coordinates.We will abbreviate the leading coefficient as The transformations between retarded and Cartesian coordinates are where these newly-appearing vectors can be found in Appendix B. The components of the HS field in retarded coordinates can be expressed as where we have used the notation Jµ(s) By introducing the symbols we may define an infinite tower of fields F α(s) on I + through the relation where N α(s) µ(s) used the same convention as (4.5).Similar to (4.2), we will always denote α(s) .(4.8) Combining (4.4) with (4.7), we find Using the identities in Appendix B, the fall-off conditions (4.1) are transformed to where the indices α may be chosen as u or r.Note that when m = s, the fall-off condition for the totally angular components is which agrees with the lower spin cases (s = 0, 1, 2).

Asymptotic expansions of gauge conditions and EOM
As has been mentioned, we may impose the following gauge conditions for free HS gauge theory without sources.In retarded coordinates, the third condition leads to Moreover, the traceless condition (3.19) becomes and it follows that This is the traceless condition on the sphere S 2 which is equivalent to The transverse condition (3.25) is where Using the identities which are shown in Appendix B, we find By multiplying the inverse tensor N µ(s−1) β(s−1) , the above equation becomes 1.For β(s − 1) = u(s − 1), we find The components F (k) u(s) are completely fixed by except for k = 2.When k = 2, we have Cu(s−1) = 0 (4.23) and F u(s) is free.
Therefore, at least for k = 1, all the components like F uα(s−1) are either zero or determined by the symmetric and traceless one F A(s) .
Asymptotic equation of motion.We still need to solve the EOM (3.15).From the identity we find This leads to an infinite tower of equations for the boundary fields It is obvious that there is no dynamical equation for the mode with k = 1 while all the descendants with k ≥ 2 are determined through the boundary equations after imposing suitable initial conditions.

Symplectic form
We can find the pre-symplectic form from the variation principle where The symplectic form can be obtained by a further variation where we have chosen a hypersurface H to evaluate the symplectic form.The symplectic form at I + is the limit where H r is the constant r slice.It follows that the fundamental commutators are where the function and the rank 2s tensor X A(s)B(s) is constructed by where It should be symmetric and traceless among the indices of the same letter5 The explicit form of X A(s)B(s) is6 with the coefficients a(p, q; s) being The commutators (4.32) can also be derived from canonical quantization which we have checked in Appendix E. After defining the vacuum |0⟩ through the annihilation operator in the boundary theory, we obtain the correlation functions where the function β(u − u ′ ) is defined by 5 Quantum flux operators For any conserved current j µ , one may construct the corresponding flux across a hypersurface H through the formula To find the Poincaré fluxes, the conserved current should be chosen as where T µν is the stress tensor of the theory and ξ is the Killing vectors of Minkowski spacetime.
To discuss the fluxes radiated to I + , we may choose constant r slices H r in retarded coordinates and then take the limit r → ∞ while keeping the retarded time u finite lim + = lim r→∞, u finite . (5. 3) It follows that the Poincaré fluxes at I + are We may read out the flux density operators from the fluxes arrived at I + per unit time and per unit solid angle.The quantum flux operators are the (generalized) Fourier transformation of the normal-ordered flux density operators.However, the definition of the stress tensor in HS theories is rather subtle.The conserved gauge invariant Bel-Robinson tensor [71,72], a direct generalization of the canonical stress tensor, is not the quantity we sought for s ≥ 2 since it has 2s derivatives.Though there are various discussions on the gauge invariant conserved currents in the literature [73][74][75][76][77], it is believed [78] that there is no gauge invariant stress tensor for s ≥ 2 due to the no-go theorem of Weinberg and Witten [20].However, there are gauge non-invariant conserved currents, akin to the Landau-Lifshitz pseudotensor [79] in general relativity, which give rise to the gauge invariant conserved charges [80].Nevertheless, we will treat the HS fields as ordinary matter and use the formula to obtain the "stress tensor".It turns out that this "stress tensor" leads to reasonable flux operators at I + .

Fluxes
Substituting the Fronsdal action into (5.5),we find With the conditions (3.14) and (3.19),only the first two terms in the Lagrangian density contribute to the stress tensor (5.7) The stress tensor can be expanded asymptotically near where the first few orders are ) where and the explicit form of X 1 , X ρσ as well as X BC are not important in this work.For more details on the calculation, we refer to Appendix C. We may compute the Poincaré fluxes generated by Killing vectors ξ For spacetime translation generator labeled by a constant vector c µ we find energy and momentum fluxes For the Lorentz transformation generator with ω µν being a constant antisymmetric tensor, the angular momentum and center-of-mass fluxes are At the second line, we decompose t (3) ρσ as (5.9b).The total derivative term containing X 2 has no contribution after integration by parts.The terms proportional to n ρ n σ or Y B (ρ Y C σ) are also vanishing due to the identities Using the relation the angular momentum and center-of-mass fluxes become (5.18) From the Poincaré fluxes, we find the following two flux density operators The tensor P AB(s)CD(s) is doubly symmetric traceless ) and can be obtained from the following tensor We have discussed this tensor extensively in Appendix D. We have added the normal ordering symbol : • • • : to remove the annihilation operators to the right-hand side of the creation operators.Similar to the lower spin cases, two smeared quantum flux operators can be defined as ) where the function f and vector Y A can be time and angle-dependent.

Supertranslations and superrotations
The commutators between the quantum flux operators (5.22) and the fundamental field where (5.24) The rank 2s + 2 tensor ρ AB(s)CD(s) is After integration by part, the quantum flux operator M Y can be rewritten as When the test functions f and Y A are time-independent, the quantum flux operators can be interpreted as supertranslation and superrotation generators.In the literature, the supertranslation and superrotation vectors ξ f,Y are expanded as in asymptotically flat spacetime.The Lie derivative of the spin s field along the direction of ξ f,Y is (5.28) We can read out the variations of the fundamental field under supertranslation and superrotation from the leading order of the components f A(s) as For the supertranslation of the field F A(s) , we find (5.30) We conclude that the quantum flux operator iT f is the generator of supertranslation for f being time-independent.For the superrotation of the field F A(s) , we should replace the variation (5.29b) induced by Lie derivative with the covariant variation [14,15] where the connection is a symmetric traceless tensor After some algebra, we find for Y A being time-independent.In this case, after subtracting a term related to supertranslation, the quantum flux operator iM Y should be regarded as the generator of superrotation.As a consistency check, one can show that T f and M Y may also be derived from the Hamilton equation δH ξ = i ξ Ω using the above variations.
In [13][14][15], the supertranslation and superrotation generators have been extended through quantum flux operators by including the time dependencies for the functions f and vectors Y .However, closing the algebra requires Ẏ = 0 and then we realize the Carrollian diffeomorphism (intertwined with super-duality transformation), which will be shown in the next subsection for the higher spin theory.It has also been extended in general dimensions and general null hypersurfaces in [16].

The algebra among flux operators
Now it is straightforward to compute the commutators for the quantum flux operators (5.34f) The results are quite similar to the lower spin cases.We will discuss these commutators term by term.
1. New local operators.The operator O g is where the rank 2s tensor Q A(s)B(s) is doubly symmetric traceless It can be obtained from the tensor ϵ This operator is the helicity flux operator associated with HS duality transformation which will be discussed in the next section.The other new operator Q h is defined as Its commutator with the fundamental field F A(s) is non-local and we do not find a physical interpretation for this operator.Therefore, we will not pay more attention to it in the following.
2. The central terms come from two-point functions for the quantum flux operators where (s) (5.40) The identity operator I f is defined by (5.41) The divergence of the Dirac delta function δ (2) (0) has been regularized to 1 12π using the Riemann zeta function or heat kernel method [16].
3. Non-local terms.The non-local terms are ) Here the tensor ∆ A(s) (g; F ; u) is a shorthand of ∆ A(s) (g; F ; u, Ω) and one should distinguish it from ∆ A(s) (Y ; F ; u, Ω) which is the superrotation variation of the fundamental field F A(s) .Actually, it is defined as which relates to the commutator 4. There is a closed algebra for Ẏ = ġ = 0 which is similar to the intertwined algebra in the lower spin cases This algebra is one of the main results of this paper.The spin s on the right-hand side of (5.45c) can be absorbed into the definition of O g and the resulting algebra is isomorphic to each other for s ̸ = 0.

Duality transformation
In this section, we will confirm that the operator O g is the helicity flux operator associated with special super-duality transformation.
Curvature tensor.For a HS field f µ(s) , we may define a curvature tensor [52] where the tensor δ αβ µν is Due to the antisymmetric property of δ αβ µν the curvature tensor is antisymmetric under the exchange of indices µ i and It is also invariant under the exchange of any pair of indices (µ i ν i ) and (µ j ν j ) The cyclic identity and the Bianchi identity are also satisfied similar to the Riemann tensor.The Fronsdal equation is equivalent to the vanishing of the "Ricci" tensor The dual of the curvature tensor is defined through the Levi-Civita tensor and has the same symmetry as the curvature tensor.It also obeys the Bianchi identity and satisfies the equation of motion Duality transformation and the corresponding flux.The duality transformation is a rotation between the curvature tensor and its dual ) with φ a constant angle.We may introduce a dual Fronsdal field f µ(s) which has the same symmetry as the Fronsdal field and relate it to the dual curvature tensor Thus the duality transformation may be induced by rotating the fields f and f whose infinitesimal transformations are with ϵ a small positive parameter.
Similar to the vector and gravitational cases, we introduce a symmetric Fronsdal action There is a parallel dual gauge transformation generated by a symmetric traceless tensor ξ µ(s−1) From Noether's theorem, we can find a conserved current associated with the global duality transformation In the last step, we have omitted the constant parameter ϵ.We may expand the dual Fronsdal field as near I + and impose the gauge fixing conditions Then the helicity flux which radiates to We can read out the helicity density operator and construct the helicity flux operator This operator is exactly the same as (5.35).According to the terminology of [15], it becomes the generator of duality transformation for g = const.and generates special super-duality transformation when g = g(Ω).
Why helicity flux?Now let us show why we call O g helicity flux operator by substituting the mode expansion of the fundamental field in Appendix E into O g .We focus on the special case g = 1 where the tensor Equivalently, it is constructed from The next step is using the bulk creation and annihilation operators to express the boundary ones (seeing (E.11)), and we obtain where σ 3 is the third Pauli matrix.Therefore, we find where n R/L,k = b † R/L,k b R/L,k is the particle number with right/left-hand helicity.Therefore, O g=1 is the difference between the numbers of particles with right-hand and left-hand helicity.

Discussion and conclusion
In this paper, we have reduced the bosonic Fronsdal theory in Minkowski spacetime to future null infinity I + .The boundary HS theory is characterized by the fundamental field F A(s) with a non-trivial symplectic form.All the descendants are determined by the fundamental field by the boundary constraint equations up to initial data.This extends the lower spin Carrollian field theories to general spin s.The symmetry algebra (5.45), which is formed by extending Poincaré and helicity flux operators, shows the same structure as the ones in the lower spin theories.All the flux operators are quadratic in the fundamental fields and could be interpreted as generators of supertranslation, superrotation and super-duality transformation, respectively.The super-duality transformation is the angle-dependent extension of the HS duality transformation (6.15) at the null boundary.In Table 1, we list the correspondences between the bulk global symmetry transformations and the boundary local transformations.These results lead us to the conjecture that each bulk global symmetry transformation may extend to a boundary local symmetry transformation at the null hypersurfaces.These local symmetry transformations are related to the radiative flux operators from bulk to boundary.It would be interesting to check this conjecture in the future.There are still many open questions to explore.
• Further extension of the Carrollian diffeomorphism.There are HS extensions of BMS symmetry in the literature [22,24,81,82] where the supertranslation and superrotation are large HS gauge transformations.The HS BMS algebra has been extended further for Carrollian conformal scalar theory [25] which is expected to be dual to a non-trivial interacting HS theory in the bulk [66].On the other hand, we work out the quantum flux operators following from Carrollian diffeomorphism which relates to spacetime geometry and differs from the ones concerning HS gauge fields.It would be interesting to see whether it is consistent to combine HS supertranslation and superrotation with Carrollian diffeomorphism.
• General null hypersurfaces.The symmetry algebra found in this work should be valid for general null hypersurface, as has been shown in [16] for scalar theory.The general null hypersurface is intriguing since one may consider massive or non-flat spacetime HS theories.
• Super-duality transformation.As has been mentioned in the introduction, duality transformations are found in various gravitational and gauge theories.It would be better to discuss their associated super-duality transformations on null boundaries.Besides, it is rather interesting to discuss the physical origin of super-duality transformation and its various consequences.
Acknowledgments.The work of J.L. is supported by NSFC Grant No. 12005069.

A Number of independent components
In this appendix, we will review the number of independent components for a symmetric tensor in d dimensions.The results can be found in any book on the representation of Lie groups and we use the review reference [83].For a d-dimensional vector space V , the symmetric tensors of rank s form a vector space Sym s V .The number of independent degrees of freedom is equal to the dimension of the space dim(Sym s V ).The symmetric tensor forms an irreducible representation of the general linear group GL(d, R) and corresponds to the Young diagram with one row of length s as shown in Figure 1.
The dimension of any irreducible representation V λ of GL(d, R) associated with Young diagram Young diagram for a rank s symmetric tensor where (i, j) denotes the box in the i-th row and j-th column in the Young diagram, as shown in Figure 2. The product is over all boxes in the diagram and the hook length is the number of squares directly below or to the right of the square (i, j), counting itself only once.For the symmetric representation Sym s V , there is only one row with s boxes.Therefore, the dimension dim(Sym

B Identities involving coordinate transformation
In the context, we defined the four vectors in Minkowski spacetime as follows where n i is the normal vector of the unit sphere S 2 .The vectors Y A µ are related to the first three vectors by These vectors satisfy various identities which are collected in [16].In this appendix, we can derive more identities associated with N α µ and N µ α in the following and as well as

C Asymptotic expansion of stress tensor near I +
The partial derivatives of the HS gauge field are Therefore, we find the following quadratic terms consisting of the stress tensor D Doubly symmetric traceless tensor on S 2 We will study the doubly symmetric traceless tensors X A(s)B(s) , Q A(s)B(s) and P AB(s)CD(s) in this appendix.
The trace-free representation of the fully symmetric, rank k tensor T a(k) 0 is given by the formula [84,85] in three dimensions and [86] in general dimensions where and η a 1 a 2 is the metric of the manifold.Note that the formula can be simplified to In our case, i.e., d = 2, k = s, the trace-free part of a fully symmetric, rank s tensor T where For later convenience, we extend the definition of a(p; s) to p = −1 with In [86], this is checked up to rank 8 by computer.It may be proved by noticing the identity The coefficients a(p; s), b(p; s) and c(p; s) satisfy the identity Note that we have used a(−1; s) = 0 in the above equation.Therefore, the trace vanishes

D.1 Doubly symmetric tensor X A(s)B(s)
Now we will prove the formula (4.37) in the context.Introducing the notation this is obtained by taking traces p and q times for the indices As and Bs, respectively.When p = q = 0, we have We use the vielbeins e A â to decompose the metric γ AB as and thus It follows that In the first step, the indices A 1 • • • A s are symmetrized using the permutation group S s .In the second step, we use the formula (D.13).In the last step, we define the symmetric tensor Therefore, the indices B 1 • • • B s is symmetrized automatically.We may rewrite The p-th trace of We could find the following product (D.20) Then the trace-free part of the tensor a(p; s)a(q; s)γ with the coefficients a(p, q; s) being As a consistency check, we will list the tensors X A(s)B(s) for several cases of lower spin in the following.
1. s = 2.The tensor whose traces are With the formula (D.21), we find and its various traces are Therefore, the doubly symmetric traceless tensor (D.28) 3. s = 4, the symmetric tensor and its various traces are The doubly symmetric traceless tensor (D.31)

D.2 Other doubly symmetric traceless tensors and related identities
In (5.20), we also defined a doubly symmetric traceless tensor P AB(s)CD(s) .Using the identity we find There are various identities associated with the tensor X A(s)B(s) , Q A(s)B(s) and P AB(s)CD(s) .
1. To calculate the commutator between T f and the fundamental field F A(s) , we need the identity 2. To compute the commutator between M Y and the fundamental field F A(s) , we need the identity 3. To calculate the commutator between O g and the fundamental field F A(s) , we need the identities  The terms with the second derivative of F are The terms with only the first derivative of F are The terms linear in F are = terms with ∇∇Y or ∇∇Z + terms with ∇Y ∇Z (D. 46) where the first part can be turned into commutators We have used the Fierz identity Therefore, we finish the proof of the identity (D.42).

To compute the central charge in the commutator
9. To compute the central charge C M O (Y, g), we need the identity which follows from the identities

D.3 Tensors and identities in stereographic project coordinates
The previous identities may be checked in stereographic project coordinates.The non-vanishing components of the doubly symmetric traceless tensors in this coordinate system are For example, the square of X A(s)B(s) can be found to be One can also use the coordinate transformation to find the doubly symmetric traceless tensors, e.g., ∂z(s) ∂θ B(s) , (D.56a) In terms of the projective stereographic coordinates, the flux density operators are simplified greatly

Mode expansion
We can also use the mode expansion to quantize the fundamental field.After imposing the De Donder gauge, the EOM becomes a wave equation whose solution can be expanded in terms of plane waves where ε α µ(s) (k) is the polarization tensor.Here the creation and annihilation operators satisfy the canonical commutator while other commutators vanish.One can choose appropriate polarization tensors such that they obey the following completeness relation where X µ(s)ν(s) is the doubly symmetric traceless part of Here γ µν has been defined as In the context, our polarization tensor satisfies which imply the corresponding properties of (E.3).The property of being symmetric and traceless is obvious due to the construction of X µ(s)ν(s) , while the others are satisfied by the definition of γ µν , namely k µ γ µν = 0 and As in the context, we impose the fall-off which leads to Converting to retarded frame, we obtain It is straightforward to compute the commutation relation between boundary creation and annihilation operators Now we are prepared to calculate the fundamental commutator which agrees with our previous result from boundary symplectic form.
We can also use the mode expansion to derive the antipodal matching conditions up to the first two orders, where Ω P = (π − θ, π + ϕ) is the antipodal point of Ω = (θ, ϕ) on the sphere, and + (−) denotes fields at I + (I − ).This result has also been checked using Green's function for retarded and advanced solutions of the wave equation with source.

E.2 Polarization tensors
In this subsection, we discuss the polarization tensors in HS theory.
Spin one and special momentum.For simplicity, we consider the case of s = 1, and take a special momentum k µ = |k|(1, 0, 0, 1) which is followed by (E.16) We need the polarization vectors to satisfy the orthogonality and completeness relations A natural choice is and thus agree with (6.29).
General momentum.For a general momentum k µ , the construction of the polarization vectors may be rather complicated.However, we find that the properties that they need to satisfy happen to be the ones of Y A µ , namely which is not the last property (E.21) superficially.However, one can combine the polarization vectors to get General spin.The key point to derive the HS polarization tensors is noting that (E.30) can be rewritten as where we have defined which can also be interpreted as a Hodge dual One can easily find and its complex conjugate Ȳµ , which agree with the expression in the literature, such as [87].Now we can construct the polarization tensors for the HS theory Ȳµ(s) .(E.36b) A nice property is that these expressions are automatically symmetric and traceless, since we have Now we need to check the orthogonality and completeness relations, as well as (6.29).The orthogonality relation is straightforward For s = 2, we find due to the relations (E.40).Then the overall coefficient is easily to be determined.
At last, we can check where we need to use and thus 48) 33) and the doubly symmetric traceless tensor becomesP AB(s)CD(s) = γ AC X B(s)D(s) − sϵ AC Q B(s)D(s) .(D.34)

47 )
Utilizing the Bianchi identity R A[BCD] = 0, we find that the above results are canceled by (D.44).With the identity P B(s) GE(s)H P DB(s)CA(s) = P B(s) DE(s)C P GB(s)HA(s) , (D.48) the terms with ∇Y and ∇Z are terms with ∇Y and ∇Z

Table 1 :
Bulk global transformations are extended to boundary local transformations We work in a representation where the particles have either right-hand or left-hand helicity.What follows is