Higher spin de Sitter quasinormal modes

We construct higher spin quasinormal modes algebraically in $D$-dimensional de Sitter spacetime using the ambient space formalism. The quasinormal modes fall into two nonunitary lowest-weight representations of $\mathfrak{so}(1, D)$. From a local QFT point of view, the lowest-weight quasinormal modes of massless higher spin fields are produced by gauge-invariant boundary conserved currents and boundary higher-spin Weyl tensors inserted at the southern pole of the past boundary. We also show that the quasinormal spectrum of a massless/massive spin-$s$ field is precisely encoded in the Harish-Chandra character corresponding to the unitary massless/massive spin-$s$ $\text{SO}(1, D)$ representation.


Introduction
In this paper, we present an algebraic approach to the construction of higher spin quasinormal modes in de Sitter spacetime (dS). In the framework of general relativity (GR), quasinormal modes can be defined as the damped modes of some perturbation in a classical gravitational background with a horizon, like black holes and dS. Astrophysically, they are important because the least damped gravitational quasinormal mode of a Schwarzschild black hole is detected and measured by LIGO through gravitational waves emitted during the so-called "ringdown" phase [1]. The measured value of gravitational quasinormal frequency can be used to test GR which predicts that the spin and mass of a black hole completely fix gravitational quasinormal frequencies.
One standard method of finding quasinormal modes in a generic background with a horizon is solving the equation of motion for a perturbation, in most cases numerically, and then imposing in-falling boundary condition at the horizon [2,3,4,5]. In the static patch of de Sitter spacetime, c.f. (3.2), by separation of variables the radial parts of quasinormal modes are found to satisfy hypergeometric functions and hence can be solved analytically (see [6,8,7] for a summary and derivation of the analytical results associated to scalars, Dirac spinors, Maxwell fields and linearized gravity in any dimensions). The underlying reason for the existence of these analytical solutions is the large dS isometry group which organizes quasinormal modes according to certain representation structure.
Such a representation structure was first discovered in [9,10] for a massive scalar field with mass m 2 = ∆(3 − ∆), 0 < ∆ < 3 1 in dS 4 . In this case, the quasinormal modes of the scalar field comprise two (non-unitary) lowest-weight representations of the isometry algebra so (1,4), which is also the conformal algebra of R 3 . More explicitly, the authors built two lowest-weight/primary quasinormal modes of quasinormal frequency iω = ∆ and iω =∆ = 3−∆ respectively, as the two leading asymptotic behaviors of vacuum-to-vacuum bulk two-point function when one point pushed to the northern pole on the future sphere.
Upon each primary quasinormal mode, an infinite tower of quasinormal modes can be generated as so(1, 4)-descendants and the scaling dimension of every descendant can be identified as (i × quasinormal frequency). The two towers of quasinormal modes together span the whole scalar quasinormal spectrum. These results were later reformulated by [11] in the ambient space formalism and generalized to massive vector fields and Dirac spinors in the same paper. We'll call the way of constructing quasinormal modes using the dS isometry group as in [9,10,11] the algebraic approach.
The separation of variables method in [6,7] is increasingly cumbersome when applied to higher spin fields due to the rapidly increasing number of tensor structures. So in this paper we will focus on generalizing the algebraic approach to construct quasinormal modes of higher spin fields, which are formulated in the ambient space (see section 2 for a review about the ambient space formalism). In section 3, we first review the algebraic construction of quasinormal modes of a scalar field ϕ of mass m in dS d+1 using the ambient 1 The representation carried by such a scalar field is in the (scalar) complementary series. space formalism. The quasinormal spectrum found in this way can be packaged into a "quasinormal character", cf. (5.1) where the sum runs over all quasinormal frequencies and d ω is the degeneracy of quasinormal modes with frequency ω. We show that the quasinormal character of ϕ is given by where ∆ = d 2 + d 2 4 − m 2 is the scaling dimension of ϕ. According to [12,13,14], χ QN ϕ (q) is the Harish-Chandra SO(1, d + 1) character corresponding to the unitary (scalar) principal series when m > d 2 and the unitary (scalar) complementary series when 0 < m ≤ d 2 . The (∆ ↔∆) symmetry in (1.2) manifests the two towers of quasinormal modes. The generalization of the algebraic construction to massive higher spin fields is straightforward.
The only difference is that the two primary quasinormal modes have spin degeneracy. In this case, the quasinormal character is given by eq. (5.4). However, the generalization to the massless higher spin case is quite nontrivial because gauge symmetry significantly reshapes quasinormal spectrums compared to the massive case. For example, the naive (∆ ↔∆) symmetry would lead to growing modes instead of damped modes because∆ = 2 − s < 0 when s ≥ 3. Moreover the symmetry disagrees with the result of [7] for s = 1, 2.
On the representation side, the underlying reason for the difficulty in generalization is that the massless higher spin fields are in the exceptional series for d ≥ 4 and in the discrete series for d = 3 while generic massive fields are in the principal series or the complementary series 2 . In section 3.3, we'll discuss the subtleties associated to gauge symmetry in more detail and explain how to take into account gauge symmetry properly while constructing physical quasinormal modes. In particular, the two-tower structure still holds and the two primary quasinormal modes are given by eq.(3.37) and eq. (3.39). In addition, in section 4, we argue that the two primary quasinormal modes are produced by insertions of boundary gauge-invariant conserved currents (of scaling dimension d + s − 2) and boundary higherspin Weyl tensors (of scaling dimension 2) at the southern pole of the past sphere (see fig.   1.1). Other quasinormal modes are sourced by the descendants of these two operators.
Based on the algebraic construction described in section 3.3, we extract the physical quasinormal spectrum of massless higher spin fields in section 5 and compute the corre- 2 We exclude the partially massless fields while talking about massive fields. sponding quasinormal character. Here, we list some examples at d = 3, 4, 5 (see eq. (5.14) for a general expression working in any d): These quasinormal characters coincide with the original computation of Harish-Chandra SO(1, d + 1) characters in [15,14] and the characters appearing in the one-loop path integral of massless higher spin fields on S d+1 , the Wick rotation of dS d+1 [16]. Therefore, the quasinormal characters, which are defined in a pure physics setup, connect nonunitary lowest-weight representations of so(1, d + 1) and the unitary representations of SO(1, d + 1).
In addition, in the appendix B, we recover the quasinormal spectrum of Maxwell theory and linearized gravity in [7] by using χ QN 1 (q) and χ QN 2 (q). When d ≥ 4, the expansion of quasinormal character χ QN s (q) always starts from a q 2 term because it corresponds to a boundary higher-spin Weyl tensor insertion. When d = 3, χ QN s (q) starts from q 1+s since the spin-s Weyl tensor, which vanishes identically on the 3dimensional boundary, gets replaced by the co-called Cotton tensor [17,18,19,20], which involves (2s + 1) derivatives on the boundary gauge field of scaling dimension 2 − s.

Ambient space formalism for fields in de Sitter
In this section, we review ambient space formalism for higher spin fields in (d + 1)dimensional de Sitter spacetime, which is realized as a hypersurface η AB X A X B = 1, η AB = (−, +, · · · , +) (2.1) in the ambient space R 1,d+1 , where A = 0, 1, · · · , d + 1. Local intrinsic coordinates y µ are defined through an embedding X A (y) that satisfies (2.1) and such an embedding induces a local metric ds 2 = g µν dy µ dy ν on dS d+1 . To gain some intuitions about the ambient space description in field theory, let's consider a free scalar field ϕ(y), defined in the local coordinates y µ , of mass m 2 = ∆(d − ∆) and satisfying equation of motion: with ∇ 2 being the scalar Laplacian. This scalar field ϕ admits a unique extension φ(X) to the ambient space such that where κ is an arbitrary constant. Define radial coordinate R = √ X 2 and hence any point in the ambient space can be parameterized by (R, y µ ) via a dS foliation. In terms of the radial coordinate, the extension condition (2.2) can be rephrased as φ(X) = φ(R, y µ ) = R −κ ϕ(y) and the ambient space Laplacian can be expressed as ∂ 2 , which together with the equation of motion of ϕ, yields In particular, if we choose κ to be ∆ or∆, φ(X) becomes a harmonic function in the ambient space. Altogether, the scalar field ϕ(y) in dS d+1 of scaling dimension ∆ is equivalent to a homogeneous harmonic function φ(X) in the ambient space R 1,d+1 , i.e.
where κ = ∆ or κ =∆. The obvious technical advantage of this ambient space description is replacing the cumbersome covariant derivative ∇ µ by the simple ordinary derivative ∂ X A .
Such a simplification is more crucial when we deal with higher spin fields. One can find a very good review about the ambient space formalism for spinning fields in AdS in [21,22].
Here we present an adapted version of the ambient space description in dS.

Higher spin fields in ambient space formalism
The totally symmetric transverse (on-shell) spin-s field ϕ µ 1 ...µs (y) of scaling dimension ∆ in dS d+1 is represented uniquely in ambient space by the symmetric tensor φ A 1 ...As (X), satisfying the following equations: • Tangentiality to surfaces of constant R = √ X 2 : • The homogeneity condition: A convenient choice is κ = ∆ or κ =∆ because, as we've seen in the scalar case, it yields the simplest equation of motion as follows: • The Casimir condition, i.e. equation of motion • The transverse condition: • The traceless condition: where we've used the generating function φ s (X, U ) ≡ 1 s! φ A 1 ···As (X)U A 1 · · · U As with U A being a constant auxiliary vector. The first two conditions ensure that φ s (X, U ) is the unique uplift of ϕ µ 1 ···µs that satisfies (2.5) and the last three conditions are equivalent to the Fierz-Pauli system: The Casimir condition : The transverse condition : ∇ µ 1 ϕ µ 1 ···µs = 0 (2.12) The traceless condition : g µ 1 µ 2 ϕ µ 1 ···µs = 0 (2.13) In the remaining part of the paper, we'll call (2.6)-(2.7) the uplift conditions and (2.8)-(2.10) the Fierz-Pauli conditions.
When ϕ µ 1 ···µs is a massless spin-s bulk field, the uplift conditions and Fierz-Pauli conditions have a gauge symmetry, with the gauge transformation takes the following simple form if we choose κ = 2 − s =∆ in eq. (2.7): where ξ s−1 satisfies • Tangentiality to surfaces of constant R = √ X 2 : • The homogeneity condition: • The Casimir condition: • The transverse condition: • The traceless condition: In the intrinsic coordinate language, this set of equations implies that ξ s−1 is a transverse lead to the conformal algebra of R d . Here we list some nontrivial commutators of the conformal algebra SO(1, d + 1) acts linearly on fields in the ambient space R 1,d+1 . In particular, the generators L AB are realized as linear differential operators in both X and U : where the first term corresponds to orbital angular momentum and the second term represents spin angular momentum. The action of M ij , P i , K i , D induced by (2.23) is where we've used lightcone coordinates X ± ≡ X 0 ± X d+1 and U ± ≡ U 0 ± U d+1 .
With a little algebra, one can show that all L AB commute with the following set of differential operators: The first five operators define the uplift conditions and Fierz-Pauli conditions, cf. (2.6)-(2.10), and hence the commutation relations imply that the on-shell bulk fields carry representations of so(1, d + 1). The last operator defines the gauge transformation of a massless higher spin field and therefore one implication of [L AB , U · ∂ X ] = 0 is that the descendants of a pure gauge mode are also pure gauge. This observation is crucial when we construct quasinormal modes for massless higher spin fields.

Algebraic construction of quasinormal modes
The southern static patch of de Sitter spacetime, which corresponds to the region denoted by "S" in the fig. (1.1), has coordinates and shows a manifest spherical horizon at r = 1 or ρ = ∞ in its metric where r = tanh ρ and dΩ 2 = h ab dϑ a dϑ b is the standard metric on S d−1 (ϑ a are spherical coordinates on S d−1 ). The traditional analytical approach to quasinormal requires solving the equation of motion in bulk and imposing in-falling boundary condition, i.e. e −iω QN (t−ρ) , for the leading asymptotic behavior near the horizon at ρ = ∞.
An algebraic method of solving quasinormal modes was first used for scalar fields in dS 4 in [9,10]. In particular, the authors found that all the quasinormal modes fall into two lowest-weight representations of the conformal algebra so(1, d + 1). Therefore, it suffices to find the two lowest-weight/primary quasinormal modes, which are solutions to the equation of motion and are annihilated by K i , and the rest of the quasinormal spectrum can be generated as descendants of them. In this section, we will first reformulate this scalar story using the ambient space formalism and then generalize it to higher spin fields.

Scalar fields
Let ϕ(X) be a free scalar field of mass m 2 = ∆(d − ∆) > 0 in dS d+1 . By construction, the equation of motion (∇ 2 − m 2 )ϕ = 0 is satisfied by the boundary-to-bulk propagators, which in ambient space take the following form: where ξ A is a constant null vector in R 1,d+1 representing a point on the future/past boundary of dS d+1 . Treating α ∆ and β ∆ as mode functions in X A , they are primary with respect to the conformal algebra so(1, d + 1) if X · ξ = X + , because K i only involves derivatives ∂ X i and ∂ X − while acting on scalar fields (cf. (2.24)). By choosing ξ A = (−1, 0, · · · , 0, 1) which is the southern pole of the past sphere, we obtain the two primary quasinormal modes with quasinormal frequency iω α = ∆ and iω β =∆ respectively. The rest quasinormal modes can be realized as descendants of α ∆ (X) and β ∆ (X). Though not explicitly spoken out in [9,10], this claim actually relies on two facts: (a) P i preserves the equation of motion and (b) P i preserves the in-falling boundary condition near horizon. The former is obvious as we've seen at the end of last section that the SO(1, d + 1) action preserves uplift conditions and Pauli-Fierz conditions. The latter holds because P i is dominated by With the quasinormal modes known, we need to figure out the corresponding frequency. This is quite straightforward in our formalism. By construction, each quasinormal mode is an eigenfunction of the dilatation operator D, which is just −∂ t in the static patch. Using the in-falling boundary condition e −iω QN (t−ρ) near horizon, we can identify the scaling dimension, i.e. eigenvalue with respect to D, as i × (quasinormal frequency ω QN ). For example, a descendant of α ∆ at level n is a quasinormal mode of frequency ω QN = −i(∆ + n) and similarly for β ∆ .
To end the discussion about scalar quasinormal modes, let's compare our construction with the known result in literature. For example, in [7], the scalar quasinormal modes in dS d+1 are found to be where Y σ (Ω) denote spherical harmonics on S d−1 and the quasinormal frequency ω takes the following values For fixed and n, the quasinormal modes of frequency ω ,n orω ,n have degeneracy D d , i.e.
the dimension of the spin-representation of SO(d). In particular, the two quasinormal modes corresponding to = n = 0 are where A d−1 is the area of S d−1 . Apart from the normalization constant, these two quasinormal modes are exactly the primary quasonormal modes α ∆ and β ∆ respectively. In addition to the match of the primary quasinormal modes, we can also show that the algebraic construction reproduces the quasinormal spectrum (3.6). Define P = √ P i P i and P i = P −1 P i . Then the linear independent descendants of α ∆ are of the form P 2n+ Y σ (P )α ∆ with , n ∈ N. For fixed and n, these are quasinormal modes corresponding to ω ,n . Similarly P 2n+ Y σ (P )β ∆ represents quasinormal modes corresponding toω ,n .

Massive higher spin fields
As in the scalar case, we need to start from a solution of the Pauli-Fierz conditions (2.8)- (2.10), subject to the tangential condition and homogeneous condition. A natural candidate is the higher spin boundary-to-bulk propagator. In AdS, the boundary-to-bulk propagator of a spin-s field with a generic scaling dimension ∆( = d + s − 2) is given by [23,24,22] K AdS where the null vector ξ ∈ R 1,d+1 represents a boundary point and the null vector Z ∈ C 1,d+1 , , which is the analogue of κ = ∆ in eq. (2.7). In AdS, this scaling property corresponds to the choice of ordinary boundary condition. In dS, on the other hand, both near-boundary fall-offs of a bulk field are dynamical and hence there are two boundaryto-bulk propagators where κ ∈ {∆,∆}. Notice that K i in (2.24) doesn't involve any derivative with respect to X + or U + . So we can obtain primary mode functions from K dS [κ,s] (X, U ; ξ, Z) by putting ξ A at the southern pole of the past sphere, i.e. ξ A = (−1, 0, · · · , 0, 1) and choosing α [∆,s] and β [∆,s] are clearly primary quasinormal modes since in-falling boundary condition near horizon naturally follows from the lack of dependence on X − and U − . Given the two primary quasinormal modes, the whole quasinormal spectrum can be generated by acting P i on them repeatedly. In this sense, the algebraic construction of quasinormal modes for massive higher spin fields is a straightforward generalization of the scalar case. Before moving to massless higher spin fields, we want to emphasize that rigorously speaking, is not one primary quasinormal mode because varying z i would yield different quasinormal modes. Indeed, α [∆,s] represents a collection of quasinormal modes with the same frequency and the vector space spanned by these quasinormal modes furnishes a spin-s representation of SO(d). But for convenience, in most part of the paper, we'll stick to the misnomer by calling, say α [∆,s] , the α-mode.

Massless higher spin fields
When ∆ hits d − 2 + s, i.e. the massless limit, K AdS [∆,s] still holds as a boundary-to-bulk propagator in the so-called de Donder gauge [23,22]. So we can extract de Sitter primary quasinormal modes, in de Donder gauge, from K AdS [∆,s] as in the massive case: and δβ (s) = U · ∂ X (· · · ).
Naively, one would expect (3.11) to be the end of story since we can generate the rest quasinormal modes as descendants of α (s) and β (s) , just as in the massive case. However, this expectation is only partially correct because, as we'll show in the following, the quasinormal spectrum is significantly affected by gauge symmetry in the massless case compared to its massive counterpart. For example, the β (s) -mode has quasinormal frequency iω = 2 − s, which would lead to an exponentially growing rather than damped behavior at future for s ≥ 3. Gauge symmetry should be the only cure for this pathological growth and indeed, we do find that the β-mode is pure gauge for any s ≥ 1: where the gauge parameter ξ s−1 can be realized as a descendant of another mode in the following sense: As a result, all the quasinormal modes in the β-tower are unphyical. The α-mode itself is not pure gauge but it has some pure-gauge descendants: where strips off z i while respecting its nullness. If we write out the indices explicitly, (3.14) means P i 1 α (s) i 1 ···is (X, U ) = 0 up to gauge transformation, which is the reminiscence of a spin-s conserved current.
In the remaining part of this section, we'll show that although the β-tower of quasinormal modes gets killed by gauge transformation, there exist a brand new tower of physical quasinormal modes.

Maxwell fields
The primary β-mode of a massless spin-1 field is β It is pure gauge and the corresponding gauge parameter takes a very special form where η 0 is given by eq. (3.13) with s = 1. Since so(1, d + 1) action commutes with gauge transformation, we can switch the order of P i and U · ∂ X in β This new expression of β can be thought as a "pure gauge" mode with the gauge parameter being U · ∂ X η 0 . (We want to emphasize that this gauge symmetry structure is completely different from the bulk gauge symmetry, which takes the form δ(· · · ) = U · ∂ X (· · · ). To distinguish it from the bulk gauge symmetry, we call it a "pseudo" gauge symmetry and its connection with the boundary gauge transformation will be discussed in section 4.2). Since β (1) i is pure gauge with respect to the pseudo gauge symmetry, the (pseudo) field strength F ij ≡ P i β vanishes identically. The vanishing of F ij signals a potential way to obtain the new physical quasinormal modes, which will be implemented step by step as follows: • First, we define a different β-modeβ From the bulk field theory point of view, it amounts to giving a mass term to the Maxwell field to break the bulk U (1) gauge symmetry.
• Then the new pseudo field strengthF ij ≡ P iβ • Stripping off the numerical factor 2(∆ − 2) and taking the limit ∆ → 2 for the remaining part, we obtain a new non-pure-gauge mode function that is antisymmetric It's straightforward to check that γ (1) ij satisfies all the requirements of being a quasinormal mode of frequency iω = 2. Written in the form of (3.19), γ (1) ij looks like a descendant of u i X + . However, this descendant structure doesn't have any physical meaning because u i X + fails to satisfy the tangentiality condition (2.6). Actually, γ (1) ij is a primary up to gauge transformation: ij is a physical primary quasinormal mode and we will call the whole Verma module built from γ (1) ij the "γ-tower" of quasinormal modes. In the framework of pseudo gauge symmetry, γ (1) ij can be thought as a U (1) field strength with u i X + being the gauge potential. As a field strength, γ ij satisfies Bianchi identity P [i γ (1) jk] = 0 which imposes a nontrivial constraint on the descendants of γ (1) ij . When d = 3, the field strength γ (1) ij is dual to a vectorγ  We'll leave the comparison with intrinsic coordinate computation of quasinormal modes of Maxwell theory to appendix A.

Linearized gravity d ≥ 4
The primary β-mode associated to a massless spin-2 field is β (2) = Φ 2 (X + ) 2 . According to eq. (3.12) and (3.13), β (2) can be alternatively expressed as where the null vectors z in β (2) are stripped off. Treating P i as an ordinary derivative, the first two terms in β (2) ij have the form of diffeomorphism transformation of (Euclidean) linearized gravity in R d . Given this pseudo diffeomorphism structure, we can naturally kill these two terms by considering the (pseudo) linearized Riemann tensor R[β (2) ] ijk , which is defined as The remaining pure trace term in β (2) ij drops out by projecting R[β (2) ] ijk to the "linearized Weyl tensor" C[β (2) ] ijk which is defined as the traceless part R[β (2) ] ijk and carries the representation of SO(d). By construction, C[β (2) ] ijk would vanish just as the U (1) field strength F ij in the spin-1 case. Due to this similarity, it's quite natural to expect the new spin-2 physical primary quasinormal mode will be produced by the same "deformation+ limiting" procedure, whose steps are listed here again for readers' convenience: (a) deform the β (2) mode by sending it toβ (2) (2) ] ijk associated toβ (2) , (c) strip off the overall factor (∆−2) in C[β (2) ] ijk and take the limit ∆ → 2 for the remaining part. To show the limiting procedure (c) more explicitly, expand, for instance, the first term in R[β (2) ]: where the convention for symmetrization is x (k P j) = x k P j + x j P k . The last term in (3.23) does not contribute to the Riemann tensor R[β (2) ] and we'll drop it henceforth. The remaining terms are proportional to ∆−2, so the limiting procedure is applicable to them where the last term drops out from Riemann tensor R[β (2) ]. Therefore the new physical quasinormal mode is schematically which has scaling dimension 2 (or quasinormal frequency iω = 2) and carries the representation of SO(d). The primariness of γ (2) ijk is proved in appendix C.
The d = 3 case is degenerate and requires a separate discussion because the 3D Weyl tensor C[β (2) ] ijk vanishes identically for arbitrary choice of ∆. So the deformation and limiting procedure used above fails to yield any quasinormal mode when d = 3. The solution to this problem is using Cotton tensor, the 3D analogue of Weyl tensor. On a 3-dimensional Riemann manifold with metric g ij , the Cotton tensor is given by [17,18] and the vanishing of Cotton tensor is the necessary and sufficient condition for the 3dimensional manifold to be conformally flat. (In spite of the abuse of notation C, it will be clear from the context that C means Weyl tensor when d ≥ 4 and means Cotton tensor when d = 3). At the linearized level, i.e. g ij = δ ij + φ ij , the Cotton tensor becomes are linearized Ricci tensor and linearized Ricci scalar respectively. In addition, the linearized Cotton tensor is actually symmetric in i, j which can be checked by contracting it with ijm where the last step is a well-known result of Bianchi identity of Riemann tensor. Since C[φ] ij is symmetric and traceless, it will be convenient to restore the null vector z In the context of quasinormal modes, we can similarly construct a pseudo Cotton tensor with ordinary derivative ∂ i in eq. (3.29) replaced by momentum operator P i Because Cotton tensor is invariant under diffeomorphism and local Weyl transformation by construction, C[φ; z] vanishes exactly when φ = β (2) . As a result, applying the deformation and limiting procedure to the Cotton tensor C[β (2) ; z] would yield a new physical primary quasinormal mode where (x ∧ u) i = ijk x j u k . Compared to higher dimensional cases, γ (2) in d = 3 is different mainly in two ways: (a) it has scaling dimension 3 because Cotton tensor involves three derivatives while Weyl tensor only involves 2, (b) it carries a spin-2 representation of SO (3) and the Bianchi identity in higher dimension becomes a "conservation" equation P i γ (2) ij = 0. Due to these two properties, the γ (2) -tower of quasinormal modes in dS 4 is isomorphic to the α (2) -tower.

Massless higher spin fields
With the spin-1 and spin-2 examples worked out explicitly, we'll continue to show that for any massless higher spin field, there exist the primary γ-mode. For a massless spin-s field, the primary β-mode given by eq. (3.12) and (3.13) can be written as β (s) Treating P i as an ordinary derivative, apart from the pure trace part, β (s) has the form of gauge transformation of d-dimensional linearized spin-s gravity. Such gauge transformation can be eliminated by using the higher spin Riemann tensor where Π ss is a projection operator ensuring R In appendix C, we show that γ (s) i 1 1 ,··· ,is s represents D d ss primary quasinormal modes of scaling dimension 2 that carry Y ss representation of SO(d), where D d ss is the dimension of the Y ss representation. (We'll use D d nm for the dimension of Y nm representation of SO(d) and D d n for the dimension of Y n representation.) In the same appendix, we also show that these primary quasinormal modes can alternatively be expressed in the following form where T ss (x, u) is a homogeneous polynomial in both x and u of degree s and satisfies (C.10). The space of such T ss carries the Y ss representation of SO(d) [27]. One obvious example of T ss is In the representation language, the example given by eq. (3.36) is actually the lowestweight state in Y ss . Therefore, we are able to generate the whole γ-tower by acting P i and M ij on the following quansinormal mode: This is a strikingly universal expression that works for any s ≥ 1 and d ≥ 4. In static patch coordinate, the nonvanishing components of (3.37) are γ (s) lw,a 1 ···as (t, r, Ω) = r 2s e −2 t (1 − r 2 ) (Ω 12 ∂ ϑ a 1 Ω 34 − Ω 34 ∂ ϑ a 1 Ω 12 ) · · · (Ω 12 ∂ ϑ as Ω 34 − Ω 34 ∂ ϑ as Ω 12 ) (3.38) where Ω 12 = Ω 1 +iΩ 2 and Ω 34 = Ω 3 +iΩ 4 . One can check that the Ω-dependent part of (3.38) is actually a divergence-free spin-s tensor harmonics on S d−1 . This is also expected from the representation side because these tensor harmonics also furnish the Y ss representation of SO(d).
For the completeness of the final result, we also give the unique lowest-weight state in the α (s) -tower here Then all the quasinormal modes are built from α

Quasinormal modes from a QFT point of view
In the previous section, we presented a pure algebraic method to construct quasinormal modes of scalars and higher spin fields. In this section, we'll provide a simple physical picture for this method from a bulk QFT point of view. In particular, we'll use scalar fields and Maxwell fields to illustrate this intuitive picture explicitly and then give a brief comment on general massless higher spin fields. Through out this section, the bulk quantum fields are defined in the southern past planar coordinate (η, y i ) of dS d+1 (the region "S"+"P" in fig. (1.1)): where η < 0 and the quasinormal modes are still defined in the southern static patch, which corresponds to y < −η in eq. (4.1).

Scalar fields
Let ϕ be a scalar field of scaling dimension ∆. Near the past boundary, it has the following asymptotic behavior The bulk two-point function of ϕ defined with respect to the Euclidean vacuum |E is given by [28] E|ϕ(X)ϕ(X )|E = Γ(∆)Γ(∆) where P = X · X . We push X to the past southern pole, i.e. y i = 0 and η → 0 − , then P is approximately − X + 2η → ∞. For P → ∞, the hypergeometric function in (4.4) has two leading asymptotic behaviors: P −∆ and P −∆ . Schematically, it means where c ∆ and c∆ are two constants. Comparing the eq. (4.2) and eq. (4.5), we find that

Maxwell fields
We want to derive the two primary quasinormal modes of Maxwell field in dS 4 using local operators. First, let's pull back α (1) i and γ (1) ij to planar patch coordinates. The α-mode: For later convenience, we perform a gauge transformation to kill the timelike component, which can be done by choosing the following gauge parameter 5 The resulting spatial part of α The γ-mode: The timelike component is automatically vanishing.
Next, we do a mode expansion for a Maxwell field A µ in the Coulomb gauge [29]: They also satisfy the following vacuum two-point functions in momentum space: Like in the scalar case, we insert the primary operator O (α) i at the southern pole of the past sphere and it produces a mode in the bulk: where the integral over k depends on the relative size of y and −η because 6 In a physical picture, the jumping at |a| = 1 reflects horizon crossing. For quasinormal modes defined in southern static patch, which has y < −η, the k-integral in (4.13) corresponds to the top case of (4.14): Up to normalization, we precisely reproduce the α Due to the extra 1 k in the integrand compared to eq.(4.13), this mode suffers from an IR divergence around k = 0. However, this divergence can be eliminated if we replace O (β) which is exactly the γ ij is indeed the curvature corresponding to the boundary gauge symmetry. Therefore, the classically pseudo gauge symmetry can be identified as the boundary gauge symmetry in quantum 6 The two cases can be uniformly treated if we give a a small positive imaginary part, i.e. a → a+i , > 0.
With this i prescription, i 2 log a+i +1 a+i −1 works for both cases. In bulk, it amounts to Wick rotating the planar coordinate time: η → e i η.

theory. In this sense, O (γ)
ij ∼ ijk B k has the interpretation as a boundary magnetic field and O (α) i ∼ E i has the interpretation as a boundary electric field, subject to the constraint ∇ · E = 0. So the quasinormal modes of Maxwell fields are produced by the electric/magnetic field operator, together with their derivatives, inserted at the past southern pole, cf. fig   (1.1).
In general, a free massless higher spin field ϕ µ 1 ···µs in a suitable gauge has the following asymptotic behavior near the past boundary where O

Quasinormal modes and SO(1, d + 1) characters
In the section 3, we describe a procedure to construct quasinormal modes of scalar fields and higher spin fields in dS d+1 . In this section, we'll extract the whole quasinormal spectrum by using this construction and show that it's related to the Harish-Chandra group character of SO(1, d + 1). To collect the information of quasinormal modes of certain field φ in a compact expression, we define a "quasinormal character" : where the sum runs over all quasinormal frequencies of φ and d ω is the degeneracy of quasinormal modes with frequency ω. Due to the representation structure of the quasinormal modes, the quasinormal character χ QN φ (q) naturally splits into two different parts, with each part involves either the α-tower or β/γ-tower of quasinormal modes. For example, let φ be a real scalar field of scaling dimension ∆. The quasinormal modes in the α-tower 7 For a bulk gauge transformation δϕ µ1···µs = ∇ (µ1 ξ µ2···µs) , the asymptotic behavior of ξ near the past boundary is ξ i1···is (η, y) ≈ (−η) d A i1···is−1 (y) + (−η) 2−2s B i1···is−1 (y). O have frequencies iω = ∆ + n, n ≥ 0 and for each n the degeneracy is d+n−1 d−1 . Thus the α-part of the quasinormal character is Similarly, the contribution of β-tower is χ QN,β ∆ (q) = q∆ (1−q) d . Altogether, we obtain the full quasinormal character of φ According to [12], q ∆ +q∆ (1−q) d is exactly the Harish-Chandra character χ HC ∆ (q) ≡ tr q D 8 for the scalar principal series, i.e. ∆ ∈ d 2 +i R and the scalar complementary series, i.e. 0 < ∆ < d. Note that the principal/complementary series condition just ensures a positive mass term for φ, which in bulk is nothing but the unitarity condition. Therefore, the quasinormal character of a unitary scalar field is same as its Harish-Chandra character. For a massive spin-s field, the story is almost the same except the α-modes and β-modes have spin degeneracy D d s , the dimension of spin-s representation of SO(d). Taking into account this spin degeneracy, we obtain the quasinormal character of a spin-s field of scaling dimension ∆, which is exactly the Harish-Chandra character for the spin-s principal series, i.e. ∆ ∈ d 2 +i R and the spin-s complementary series, i.e. 1 < ∆ < d − 1.
In the remaining part of this section, we'll compute quasinormal characters for massless higher spin fields. In this case, the α-part is easy because α (s) i 1 ···is is a conserved current in the sense of (3.14). So for iω α n = d + s − 2 + n in the α-tower, the degeneracy is is the same as the SO(2, d) character corresponding to a massless spin-s field in AdS d+1 [30,31]. However, this is far from the corresponding SO(1, d + 1) character [14,12].
In the notation of [14], the massless spin-s representation of SO(1, d + 1) is denoted by D j (α;p) , with p = 0, j = d−4 2 for even d, j = d−3 2 for odd d and α = (s, s, 0, · · · , 0) 9 . In the notation of [12], this representation falls in the exceptional series with ∆ = p = 2 and Y p = Y ss . The corresponding Harish-Chandra SO(1, d + 1) character is For 2 ≤ n ≤ d 2 , D n is the dimension of SO(d) representation Y (ss,1 m ) , obtained by adding n − 2 single-box rows to Y ss . (When d = 2r and n = r, D r is actually the dimension of −1) ). These values can be easily extended to d 2 + 1 ≤ n ≤ d − 2 by the manifest n ↔ d − n symmetry of D n .
To compare the quasinormal character χ QN s with the Harish-Chandra character χ HC s , we still need to figure out quasinormal spectrum of the γ-tower.

Maxwell field
Let's start from a Maxwell field. At level 0, i.e. iω = 2, the degeneracy is d γ ij carries the 2-form representation of SO(d). At level 1, generic descendants are of the form P k γ (1) ij , corresponding to the SO(d) representation ⊗ . The 3-form representation in this tensor product is vanishing due to Bianchi identity P [k γ (1) ij] = 0. Therefore the degeneracy of quasinormal modes with frequency iω = 3 is The descendants at level 2 are P k P γ (1) ij , corresponding to the SO(d) representation (• ⊕ ) ⊗ . Due to Bianchi identity, we would exclude terms like P P [k γ (1) ij] , that carries the ⊗ representation. However, this is overcounting because the 4-form representation in this tensor product, carried by P [ P k γ (1) ij] , vanishes automatically without using Bianchi identity. Therefore the degeneracy of quasinormal modes with frequency iω = 4 is At any level n, using the same argument, we obtain the degeneracy of quasinormal modes with frequency iω = 2 + n which leads to the γ-tower quasinormal character Combining eq.(5.5) for s = 1 and eq.(5.11), we get the full quasinormal character of a Maxwell field On the other hand, since D n reduces to d n when s = 1, the Harish-Chandra character (5.6) for s = 1 is Again, quasinormal character=Harish-Chandra character. In appendix B, we'll show that is also consistent with the spin-1 quasinormal spectrum in [7].

Higher spin fields
To check χ QN s = χ HC s for any s ≥ 2, it is easier to use a different but equivalent expression of χ HC s found in [16]: where [ ] + is a linear operator that sends q k → −q −k for k < 0 and drops the constant term while acting a Laurent series around q = 0. As a very simple example, [q −1 +1+q] + = 0.
Notice that the first term of χ HC s (q) is the same as χ QN,α s (q), so it suffices to compare the second term with χ QN,γ s (q). Expand the second term into a Taylor series around q = 0: With some simple algebra, one can show that b 0 = D d ss , b 1 = D d s+1,s +D d s,s−1 and furthermore b n satisfies the following recurrence relation Using the tensor product decomposition of Y s ⊗ Y t (assuming t ≤ s) On the quasinormal modes side, since the primary γ (s) carries Y ss representation, the degeneracy at level 0 is d γ 0 = D d ss = b 0 . At level 1, the descendants P k γ (s) Due to Bianchi identity, the threerow summand in this tensor product vanishes and the level 1 descendants only carry the Y s+1,s ⊕ Y s,s−1 representation. So the degeneracy of quasinormal frequency iω = 3 is At higher levels, we aim to derive a recurrence relation for the degeneracy d γ n . For example, at level n, the descendants are of the form P 1 · · · P n γ (s) i 1 j 1 ,··· ,isjs , where P 1 · · · P n should be understood group theoretically the symmetrized tensor product of n spin-1 representations. Compared to level (n − 2), the new representation structure is Y n ⊗Y ss where Y n corresponds to the traceless part of P 1 · · · P n and Y ss corresponds to γ (s) . Due to Bianchi identity, only two-row representations in the tensor product decomposition of Y n ⊗Y ss are nonvanishing. These two-row representations are exactly ⊕ min(n,s) =0 Y s− +n,s− , which yields d γ n − d γ n−2 = b n − b n−2 and furthermore b n = d γ n for n ≥ 0. Altogether, we can conclude χ QN s (q) = χ HC s (q) (5.20)

Conclusion and outlook
In this paper, we present an algebraic method of constructing quasinormal modes of massless higher spin fields in the southern static patch of dS d+1 using ambient space formalism.
With the action of isometry group SO(1, d + 1), the whole quasinormal spectrum can be built from two primary quasinormal modes, whose properties are summarized in the table.
representation of SO(d). The conservation law means that P i 1 α µν i 1 i 2 is pure gauge and hence should be excluded from the physical spectrum of quasinormal modes. On the other hand, the primary γ-modes γ µν i 1 j 1 ,i 2 j 2 have quasinormal frequency ω QN = −2i and degeneracy transform as Weyl tensor under SO(d). This also explains the Bianchi identity P [k γ µν i 1 j 1 ],i 2 j 2 = 0. With the higher spin quasinormal modes known, we define a quasinormal character χ QN s (q), cf. (5.1) that encodes precisely the information of quasinormal spectrum. We show that χ QN s (q) is equal to the Harish-Chandra group character χ HC s (q) of the unitary massless spin-s SO(1, d + 1) representation. In other words, the pure group theoretical object χ HC s (q) knows everything about the physical quasinormal spectrum. Our algebraic approach to quasinormal modes has some potential generalizations and applications which will be left to investigate in the future: • Construct quasinormal modes of fields carrying other unitary representations, for example partially massless fields or discrete series fields. The generalization to partially massless fields should be more or less straightforward. In particular, the construction of the primary α-modes would be the same except the conservation law being replaced by a multiply-conservation equation [32]: where t is the depth. For the primary γ-modes, the higher spin Weyl tensor used in eq. (3.34) is expected to be replaced by its partially massless counterpart which carries Y s,t+1 representation of SO(d) [33]. The discrete series case should be different because it is labelled by a maximal height Young diagram. As a result, neither of the primary quasinormal modes can be a curvature like object. This is also confirmed from the character side [12].
• Generalize the "quasinormal quantization" [9,10] to massless higher spin gauge fields in any higher dimension. It's well known that quasinormal modes are nonnormalizable with respect to the standard Klein-Gordon inner product. However, it is noticed in [9,10] that, at least for light scalar fields in dS 4 , there is the so-called "R-norm" such that the quasinormal modes become normalizable and SO(1, 4) is effectively Wick rotated to SO (2,3). Granting the existence of "R-norm" in higher dimensions for massless higher spin fields that maps SO(1, d + 1) to SO(2, d), then the γ-tower of quasinormal modes carries the [∆ = 2, Y ss ] representation of SO (2, d), that is below the unitarity bound for sufficiently large s. This simple argument seems to question the naive generalization of "R-norm".
• In this paper, we've focused on Harish-Chandra character χ R (q) = tr R q D with only the scaling operator D turned on, where R denotes some unitary irreducible representation. In general, we can also include SO(d) generators in the definition of characters: where J i = L 2i−1,2i span the Cartan algebra of SO(d) and x i are auxiliary variables.
For example, for spin-s principal series or complementary series, the full character (6.2) reads [12,14] χ HC [∆,s] (q, x) = (q ∆ + q∆)χ x Jr r denotes the SO(d) character of spin-Y representation and For massless spin-s representation, the full character originally computed in [14] is: We conjecture that the full Harish-Chandra character encodes spin content of quasinormal modes. More precisely, expand χ HC s (q, x) in terms of q and x i χ HC s (q, x) = ω, j d ω, j q iω x j 1 1 · · · x jr r , j = (j 1 , · · · , j r ) (6.7) then d ω, j is conjectured to be the degeneracy of quasinormal modes with frequency ω and spin content j. In this appendix, we show the agreement between our algebraically constructed primary quasnormal modes and their intrinsic coordinate counterparts in literature for free Maxwell fields. According to [7], the quasinormal modes of Maxwell theory can be divided into the following two types: where ϑ a are the spherical coordinates on S d−1 and Y σ are scalar spherical harmonics with σ being a collective symbol for the magnetic quantum numbers, where Y σ a are divergence-free vector spherical harmonics on S d−1 . In type I solutions, the radial function R (I) (r) is given by with the quasinormal frequency ω valued in iω I ,n = + d − 2 + 2n, iω I ,n = + 2 + 2n (A.4) In type II solutions, the radial function R (II) (r) is given by with the quasinormal frequency ω valued in iω II ,n = + d − 1 + 2n, iω II ,n = + 1 + 2n (A.6) In both type I and II, ≥ 1 and n ≥ 0. In the following, we show that the primary quasinormal modes α On the other hand, the pull-back of α Naively, A (I) µ and α i,µ look different. This is because the former is solved in a modified Feynman gauge [34] while the latter follows from boundary-to-bulk propagator in de Donder gauge, which for spin-1 field is simply the Lorenz gauge. To compare the two results, we perform a gauge transformation α (1) i,µ →α (1) i,µ = α i,µ + ∂ µ ξ i to set the t-component zero. The simplest choice of the gauge parameter is ξ i = 1 d−1 α i,t . Due to this gauge choice, the new α Since {Ω i } i and {Y 1σ } σ are just different basis for the same vector space of spherical harmonics of eigenvalue −(d − 1) with respect to ∇ 2 S d−1 , eq.(A.7) and eq. (A.9) actually represent the same set of quasinormal modes.

B Match quasinormal spectrums
In the section 5, we defined a quasinormal character χ QN for a given quasinormal spectrum {ω, d ω }, cf. (5.1). By definition, the correspondence between quasinormal characters and quasinormal spectrums is one-to-one . In this appendix, by using quasinormal characters, we show that our algebraic construction yields the same quasinormal spectrum as [7] for Maxwell fields and linearized gravity. On the algebraic side, the quasinormal character of a massless spin-s field is shown to be given by eq. (5.14). In particular, for s = 1 and s = 2, the quasinormal characters read and

Maxwell fields
In the last appendix, we've summarized the quasinormal modes of Maxwell fields computed in [7]. Here let's briefly recap the information about quasinormal frequencies: type I : iω I ,n = + d − 2 + 2n, iω I ,n = + 2 + 2n type II : iω II ,n = + d − 1 + 2n, iω II ,n = + 1 + 2n where ≥ 1 and n ≥ 0. For fixed and n, each frequency of type I quasinormal modes has degeneracy D d because labels scalar spherical harmonics while each frequency of type II quasinormal modes has degeneracy D d 1 because labels divergence-free vector spherical harmonics. So the quasinormal character associated to the spectrum (B.3) is where the first sum over simply follows from The second sum over in (B.4) can be derived using In particular, when s = 1, and hence the second sum reduces to the (B.5) type: Plugging (B.5) and (B.7) into the quasinormal character (B.4) yields which shows the agreement between our algebraic method and the traditional analytical method on the quasinormal spectrum for Maxwell theory.

Linearized gravity
Quasinormal modes of linearized gravity are divided into three categories. The three types of fluctuation can be solved simultaneously by using the so-called Ishibashi-Kodama equation [2,36,7]. The quasinormal frequencies are: Scalar type fluctuation : iω S ,n = + d − 2 + 2n, iω S ,n = + 2 + 2n Vector type fluctuation : iω V ,n = + d − 1 + 2n, iω V ,n = + 1 + 2n Tensor type fluctuation : iω T ,n = + d + 2n, iω T ,n = + 2n (B.9) where ≥ 2 and n ≥ 0. In these 3 types of fluctuations, labels scalar spherical harmonics, divergence-free vector spherical harmonics and divergence-free tensor spherical harmonics on S d−1 respectively and hence for fixed and n, each frequency has degeneracy D d , D d 1 and D d 2 respectively. Altogether, the quasinormal character associated to the spectrum (B.9) is given by where the first two series of are essentially computed in the Maxwell field case and the last series follows from eq.(B.6) with s = 2: Combine the three series of in (B.10) and we obtain which is exactly χ QN 2 (q). This computation confirms the match of quasinormal spectrum for linearized gravity. 10 C Details of γ (s) i 1 1 ,··· ,i s s The higher spin quasinormal mode γ (s) i 1 1 ,··· ,is s defined by eq. (3.34) is rather schematic. In this appendix, we will write out its explicit form and then show various properties of it. 10 When d = 3, the tensor type fluctuations doesn't exist because there is no divergence-free tensor harmonics on S 2 . However, one can still recover the quasinormal character χ QN 2 (q) for d = 3 by counting quasinormal modes in the scalar and vector type fluctuations in this case.
Near horizon γ (s) i 1 1 ,··· ,is s becomes singular because X + → 0. This singular behavior also shows the following in-going boundary condition: where we can directly read off the quasinormal frequency iω = 2.
The next task is to show that γ (s) i 1 1 ,··· ,is s is primary up to gauge transformation by using its explicit form (C.8) (we can also use (C.9) with T ss (x, u) subject to (C.10)). Acting the special conformal transformation K m on γ (s) i 1 1 ,··· ,is s , we get (dropping the normalization constant c s ) where we've replaced U ·∂ X by u·∂ x in the second term of the second line because χ where we've restored the normalization constant c s .