New conformal-like symmetry of strictly massless fermions in four-dimensional de Sitter space

We present new infinitesimal `conformal-like' symmetries for the field equations of strictly massless spin-$s \geq 3/2$ totally symmetric tensor-spinors (i.e. gauge potentials) on 4-dimensional de Sitter spacetime ($dS_{4}$). The corresponding symmetry transformations are generated by the five closed conformal Killing vectors of $dS_{4}$, but they are not conventional conformal transformations. We show that the algebra generated by the ten de Sitter (dS) symmetries and the five conformal-like symmetries closes on the conformal-like algebra $so(4,2)$ up to gauge transformations of the gauge potentials. The transformations of the gauge-invariant field strength tensor-spinors under the conformal-like symmetries are given by the product of $\gamma^{5}$ times a usual infinitesimal conformal transformation of the field strengths. Furthermore, we demonstrate that the two sets of physical mode solutions, corresponding to the two helicities $\pm s$ of the strictly massless theories, form a direct sum of Unitary Irreducible Representations (UIRs) of the conformal-like algebra. We also fill a gap in the literature by explaining how these physical modes form a direct sum of Discrete Series UIRs of the dS algebra $so(4,1)$.

A Deriving Eq. (5.2) by analytically continuing so(5) rotation generators and their matrix elements to so(4, 1) 31 A.1 Background material for representations of so (5) and Gelfand-Tsetlin patterns 31 A.2 Specialising to so (5) representations formed by tensor-spinor spherical harmonics on S 4 32 A.3 Transformation properties of tensor-spinor spherical harmonics on S 4 under so(5) 34 A. 4 Performing analytic continuation 38 B Details for the computation of the commutator (6.16) between two conformallike transformations 40 1 Introduction Four-dimensional de Sitter spacetime (dS 4 ) is believed to be a good approximation of the very early epoch of our Universe (Inflation).Also, according to recent data indicating the accelerated expansion of space [1][2][3], there is evidence to suggest that our Universe is asymptotically approaching another de Sitter phase.
The D-dimensional de Sitter spacetime (dS D ) is the maximally symmetric solution of the vacuum Einstein equations with positive cosmological constant Λ, where g µν is the de Sitter metric tensor, R µν is the Ricci tensor and R is the Ricci scalar.
In this paper, we use units in which 2Λ = (D − 1)(D − 2), while the Riemann tensor is R µνρσ = g µρ g νσ − g νρ g µσ . (1.2) De Sitter (dS) field theories are known to exhibit characteristics with no Minkowskian analogs.Two such interesting characteristics of integer-spin fields on dS D -related to the representation theory of the dS algebra so(D, 1) -are: • The existence of unitarily forbidden ranges for the mass parameters of integer-spin fields depending on both D and the spin of the fields [14,41].
(1.4)Moreover, Higuchi observed that for the following special values of the mass parameter [14,16,41]: the theory is unitary, while at the same time it enjoys a gauge symmetry.A field with mass parameter given by Eq. (1.5) is a gauge potential known as partially massless field of depth τ in the modern literature [6,8,9]. 1 The case with τ = 1 corresponds to the theory known as strictly massless.In 4 dimensions, a strictly massless field has two propagating helicity degrees of freedom ±s, while a partially massless field of depth τ has 2τ of them: (±s, ±(s − 1), ..., ±(s − τ + 1)) [6,8,9].In dS field theory, the strictly massless fields are the closest analogs of Minkowskian massless fields, while partially massless fields of depth τ > 1 have no Minkowsian counterparts.The Unitary Irreducible Representations (UIRs) of the dS algebra so(D, 1) corresponding to totally symmetric strictly/partially massless integerspin fields were first discussed in Higuchi's PhD thesis [14,16].More recent discussions concerning both totally symmetric and mixed symmetry integer-spin fields can be found in Ref. [17].
What about the representation theory of fermions on dS D ?
Unlike the integer-spin case, the representation-theoretic properties of fermionic fields on dS D are not well-studied.
Recent results.Although the phenomena of strict and partial masslessness also occur in the case of spin-s ≥ 3/2 fermionic fields on dS D [6,8,9], the study of the corresponding unitarity properties was absent from the (mathematical) physics literature for a long time.Interestingly, as the author has recently shown [18,53], four-dimensional dS space plays a distinguished role in the unitarity of strictly/partially massless (totally symmetric) tensorspinors on dS D (D ≥ 3).More specifically, the representations of the dS algebra so(D, 1), which correspond to strictly/partially massless totally symmetric tensor-spinors of spin s ≥ 3/2, are non-unitary unless D = 4 [18,31,53]. 2  In the present paper, we uncover a new group-theoretic feature of all strictly massless totally symmetric spin-s ≥ 3/2 tensor-spinors on dS 4 : these fermionic gauge potentials possess a conformal-like so(4, 2) global symmetry algebra.Moreover, we show that the mode solutions with fixed helicity, i.e. the modes forming Unitary Irreducible Representations (UIRs) of the dS algebra so(4, 1) [18], also form UIRs of the larger conformal-like so(4, 2) algebra.

List of main results and methodology
Here we give some information about our main results and investigations concerning the new conformal-like symmetries of strictly massless fermions on dS 4 .
• We present new conformal-like infinitesimal transformations (6.6) for strictly massless totally symmetric tensor-spinors on dS 4 .These new transformations are generated by conformal Killing vectors of dS 4 and they are symmetries of the field equations [Eqs.(2.15) and (2.16)], i.e. they preserve the solution space of the field equations.In this paper, by conformal Killing vectors we mean the five genuine conformal Killing vectors of dS 4 with non-vanishing divergence -see Eq. ( 6.3).
• The conformal-like transformations (6.6), together with the ten known dS transformations (2.9), generate an algebra that is isomorphic to so(4, 2).However, this conformal-like algebra closes up to field-dependent gauge transformations.
• We fill a gap in the literature by clarifying the way in which the spin-s ≥ 3/2 physical (i.e.non-gauge) mode solutions with fixed helicity form a direct sum of Discrete Series UIRs of the dS algebra so(4, 1).The modes with opposite helicity correspond to different UIRs -this is also true in the case of strictly massless totally symmetric tensors [16].(Recall that a strictly massless field has only two propagating helicities ±s corresponding to two sets of physical mode solutions with opposite helicities.) • Then, we show that the physical mode solutions also form a direct sum of UIRs of the conformal-like so(4, 2) algebra.We arrive at this result by following two basic steps (which stem from the mathematical definitions of representation-theoretic irreducibility and unitarity).First, we show that the mode solutions with fixed helicity transform among themselves under all so(4, 2) transformations (this means under the ten dS isometries (2.9), as well as the five conformal-like symmetries (6.6)).Then, we show that there is a so(4, 2)-invariant, and gauge-invariant, positive definite scalar product for each set of mode solutions with fixed helicity.
For the cases with spin s ≥ 7/2, the results were obtained in Ref. [18] motivated only by the examination of the so(D, 1) UIRs and the (mis-)match of the representation labels.The technical analysis of the representation-theoretic properties of the mode solutions with spin s ≥ 7/2 is still absent from the literature for D ̸ = 4 (the D = 4 case is studied in the present paper).Thus, if we want to be careful -as we should, if we wish to avoid the representation-theoretic confusion that appeared in the past dS literature -the results of Ref. [18] for s ≥ 7/2 may be viewed as a "suggestion" motivated by the examination of the so(D, 1) UIRs.This suggestion can be confirmed by studying the representation-theoretic properties of the spin-s ≥ 7/2 mode solutions on dSD, as in the spin-s = 3/2, 5/2 cases [31,53].This is something that we leave for future work.
• As the name suggests, our conformal-like symmetry transformations are not conventional infinitesimal conformal transformations.This is exemplified as follows.For the cases with spin s = 3/2, 5/2, by investigating the conformal-like transformations of the field strength tensor-spinors (i.e.curvatures) of the strictly massless fermions 3 , we find that these transformations correspond to the product of two transformations: an infinitesimal axial rotation (i.e.multiplication with γ 5 ) times an infinitesimal conformal transformation.For the cases with spin s ≥ 7/2, we present a (justified) conjecture concerning the expressions for the conformal-like transformations of the field-strength tensor-spinors.
We conclude this part of the Introduction with a brief literature review that is relevant to our present work.The UIRs of so(4, 1) corresponding to certain fermions on dS 4 have been also discussed in Ref. [27].The mode solutions and the Quantum Field Theory of spin-1/2 fermions on dS D have been discussed in various articles, such as Refs.[13,20,22,23,27,[33][34][35][36][37][38][39][40].The invariance of maximal-depth integerspin partially massless theories on dS 4 under conformal transformations has been investigated in Ref. [10] -however, interestingly, a representation-theoretic study [42] suggests that the associated symmetry algebra does not correspond to the conformal algebra.Discrete Series UIRs, which play a central role in the present paper, also exist in the case of the isometry group of dS 2 .Recently, operators furnishing the discrete series UIRs of so(2, 1) in BF-type gauge theories on dS 2 were constructed [54].Quantum aspects of de Sitter space have been reviewed in [58,59].

Outline, notation, and conventions
The rest of this paper is organised as follows.In Section 2, we review the basics concerning (strictly massless) tensor-spinors on dS 4 .In Section 3, we review the classification of the UIRs of the dS algebra so(4, 1).In Section 4, we discuss the (pure gauge and physical) mode solutions for strictly massless fermions of spin s ≥ 3/2 on global dS 4 .In particular, we use the method of separation of variables to express the physical mode solutions on global dS 4 in terms of tensor-spinor spherical harmonics on S 3 (these spherical harmonics are not constructed explicitly here).We also identify the analogs of the flat-space positive and negative frequency modes.In Section 5, we discuss the way in which the (positive frequency) physical modes with fixed helicity form a direct sum of Discrete Series UIRs of so(4, 1).In Section 6, we present our new conformal-like symmetry transformations and we show that the associated symmetry algebra (generated by both dS and conformal-like transformations) closes on so(4, 2) up to gauge transformations.In Section 7, we show that the physical modes that form a direct sum of so(4, 1) UIRs, also form a direct sum of so(4, 2) UIRs.In Section 8, we discuss the conformal-like transformations of the gauge invariant filed strength tensor-spinors.
There are two Appendices, A and B, in which we include technical details that were omitted in the main text.
Notation and conventions.We use the mostly plus metric sign convention for dS 4 .Lowercase Greek tensor indices refer to components with respect to the 'coordinate basis' on dS 4 .Coordinate basis tensor indices on S 3 are denoted as μ1 , μ2 , ... .Lowercase Latin tensor indices refer to components with respect to the vielbein basis.Repeated indices are summer over.We denote the symmetrisation of indices with the use of round brackets, e.g.A (µν) ≡ (A µν + A νµ )/2, and the anti-symmetrisation with the use of square brackets, e.g.A [µν] ≡ (A µν − A νµ )/2.Spinor indices are always suppressed throughout this paper.The rank of spin-s tensor-spinors on dS 4 is r (i.e.s = r + 1/2).The complex conjugate of the number z is denoted as z * .By conformal Killing vector we mean a genuine conformal Killing vector of dS 4 with non-vanishing divergence -see Eq. (6.3).
The derivative ∇ ν acts on our totally symmetric tensor-spinors as where Γ λ νµ are the Christoffel symbols, while ω νbc = ω ν[bc] = e ν a ω abc is the spin connection.We have The gamma matrices are covariantly constant, ∇ ν γ µ = 0.The commutator of covariant derivatives acting on totally symmetric tensor-spinors on dS 4 is given by (2.8)

Basics about dS symmetries of the field equations
The dS algebra is generated by the ten Killing vectors of dS 4 satisfying ∇ (µ ξ ν) = 0.The dS generators act on solutions Ψ µ 1 ...µr in terms of the spinorial generalisation of the Lie derivative -also known as the Lie-Lorentz derivative [24,28].The Lie-Lorentz derivative acts on arbitrary tensor-spinors as follows: where . The Lie-Lorentz derivative L ξ Ψ µ 1 ...µr conveniently describes the infinitesimal so(4, 1) transformation of Ψ µ 1 ...µr generated by the Killing vector ξ µ .From the properties [24]: it follows that if Ψ µ 1 ...µr is a solution of Eqs.(2.1) and (2.2), then so is L ξ Ψ µ 1 ...µr .In other words, the Lie-Lorentz derivative is a symmetry of the field equations for any value of M .It is easy to conclude that the associated symmetry algebra is isomorphic to so(4, 1) as [24] [L ξ , for any two Killing vectors ξ µ and ξ ′ µ .
The dS algebra, so(4, 1), has four non-compact generators ('dS boosts') and six compact ones ('dS rotations').The compact generators generate the so(4) rotational subalgebra of so(4, 1).For any fixed value of M , the mode solutions of the field equations (2.1) and (2.2) form an infinite-dimensional representation of so(4, 1).The eigenvalue of the quadratic Casimir for this representation is given by [18] where s = r + 1/2 ≥ 3/24 .The unitarity of the representation depends on the value of the mass parameter M [18].In this paper, we are interested in the strictly massless theories, which appear for special imaginary values of M (see Subsection 2.3) -for discussions on arbitrary values of M in any spacetime dimension see Ref. [18].
In this paper, we are interested in the equations for strictly massless fermions, i.e.Eqs.(2.1) and (2.2) with mass parameter given by [6,15] M = ±i r. (2.14) Strict masslessness occurs for either of the two signs for the mass parameter in Eq. (2.14).However, the representations of so(4, 1) corresponding to the '+' sign are equivalent to the representations corresponding to the '−' sign [18].This is easy to understand as, if while, also, γ 5 commutes with all dS transformations (2.9) [18].

Classification of the UIRs of the dS algebra
In this Section, we review the classification of the so(4, 1) UIRs in the decomposition so(4, 1) ⊃ so(4) [44,45].An irreducible representation of so(4) appears at most once in a UIR of so(4, 1) [50].
Let us recall that an irreducible representation of so( 4) is specified by the highest weight [51,52] where The numbers f 1 and f 2 are both integers or half-odd integers, while f 2 can be negative.
In global coordinates, the non-zero Christoffel symbols are where gθ i θ j and Γθ k θ i θ j are the metric tensor and the Christoffel symbols, respectively, on S 3 .The vierbein fields on dS 4 can be chosen to be: where ẽθ i i are the dreibein fields on S 3 .The non-zero components of the dS spin connection are given by where ωijk is the spin connection on S 3 .We will work with the following representation of gamma matrices on dS 4 : (j = 1, 2, 3) where the lower-dimensional gamma matrices, γ j , satisfy The fifth gamma matrix (2.5) is given by

Constructing the mode solutions of the strictly massless theories
There are two kinds of spin-(r + 1/2) TT mode solutions satisfying the strictly massless field equations [(2.15) and (2.16)] on dS 4 : • The 'physical modes' describing the propagating degrees of freedom of the theory.
• The 'pure gauge modes' describing the gauge degrees of freedom of the theory.
In this Subsection, we present some details for the construction of these mode solutions.
The mode solutions on global dS 4 can be constructed using the method of separation variables.Schematically, this means that we are looking for solutions that can be expressed as a product "function of t × function of θ 3 ".As we will see below, the functions describing the θ 3 -dependence are tensor-spinor spherical harmonics on S 3 forming UIRs of so(4).Thus, from a representation-theoretic viewpoint, the solutions on global dS 4 obtained with the method of separation of variables form so(4, 1) representations in the decomposition so(4, 1) ⊃ so(4).The method of separation of variables has been applied in Refs.[14,16,49] for integer-spin fields, in Refs.[22,23] for spin-1/2 fields and in Refs.[18,53] for spin-3/2 and spin-5/2 fields.

Physical spin-
Let us start by obtaining the physical mode solutions of Eqs.(2.15) and (2.16).We first discuss the spherical eigenmodes on S 3 that describe the spatial dependence of physical modes.Then, we discuss the time dependence of physical modes and we apply the method of separation of variables.
letting µ 1 = μ1 , µ 2 = μ2 , ..., µ r = μr , the Dirac equation (2.15) for the physical modes is expressed as where we have made use of the expressions for the Christoffel symbols, spin connection, vierbein fields and gamma matrices from Subsection 4.1.
Before proceeding to the construction of the modes, note that the physical modes on dS 4 are naturally split into two classes depending on their so(4) representation-theoretic content -i.e.depending on whether their θ 3 -dependence is given by the spherical eigenmodes (4.9) or (4.10).Let us introduce the following notation: • The physical modes with so(4) content given by ⃗ f − r [Eq.(4.12)] are denoted as Ψ (phys, −ℓ; m;k) µ 1 ...µr (t, θ 3 ).We also refer to these modes as 'physical modes with helicity −s' (recall that s = r + 1/2).

Short wavelength limit of physical modes
Using the property [57]: we find that in the limit ℓ >> The pure gauge modes of the striclty massless spin-(r + 1/2) equations [(2.15) and (2.16)] satisfy the same conditions as the restricted gauge transformations (2.17).This means that the pure gauge modes are expressed as The "gauge-function modes", λ (r, ±ℓ; m) µ 2 ...µr , are totally symmetric tensor-spinors of rank r − 1 and they satisfy Eqs.(2.18) and (2.19).As in the case of physical modes, explicit expressions for λ (r, ±ℓ; m) µ 2 ...µr can be obtained using the method of separation of variables, but they are not needed for the purposes of this paper 9 .The two labels r, ±ℓ in Eq. (4.29) are used to denote the so(4) content of each pure gauge mode; this corresponds to the so(4) highest weights with ℓ = r, r + 1, ... [the value r = r is excluded in Eq. (4.30) since it corresponds to the so(4) content of physical modes -see Eqs.(4.11) and (4.12)].The label m represents angular momentum quantum numbers corresponding to the subalgebras so(3) ⊃ so(2).The pure gauge modes must have zero norm with respect to any dS invariant scalar product and be orthogonal to all physical modes [14,16,18,49,53].Because of these properties, the pure gauge modes can be identified with zero in the solution space of the field equations (2.15) and (2.16).These properties will be demonstrated in Section 5 for a specific choice of dS invariant scalar product -see also Refs.[18,53].

The physical modes form UIRs of the dS algebra
In this Section, we explain how the 'positive frequency' physical modes (4.15) and (4.24) of the fermionic strictly massless theories form a direct sum of Discrete Series UIRs of the dS algebra so(4, 1).In order to identify the so(4, 1) UIRs formed by the mode solutions, we follow two basic steps: • Irreducibility: We identify the sets of physical modes that form irreducible representations of so (4,1).
This means that we need to study the infinitesimal dS transformations of the physical mode solutions.We show that the physical modes with fixed helicity ±s transform among themselves under all so(4, 1) transformations (up to gauge equivalence).Thus, the physical modes form a direct sum of irreducible representations -one corresponding to the helicity +s and one to −s.Moreover, it is already easy to see that pure gauge modes transform only into other pure gauge modes under infinitesimal dS transformations, as the Lie-Lorentz derivative (2.9) commutes with the operator ∇ µ + i 2 γ µ in Eq. (4.29), while also, it leaves invariant the conditions (2.18) and (2.19), which determine the restricted gauge transformations.
• Unitarity: We introduce a dS invariant and gauge-invariant scalar product that is positive definite for physical modes of fixed helicity.
With respect to this scalar product, the pure gauge modes are shown to be orthogonal to themselves, as well as to all physical modes (i.e. it is demonstrated that the pure gauge modes can be identified with zero in the solution space).Interestingly, it turns out that our scalar product is positive definite for the physical modes with helicity −s and negative definite for the physical modes with helicity +s.However, as these two sets of fixed-helicity modes form different irreducible so(4, 1) representations, we are allowed to use a different scalar product for each set.We thus redefine the scalar product for the +s modes by introducing a factor of −1, in order to achieve positivedefiniteness.This peculiarity -i.e.having a different positive definite scalar product for physical modes with different helicity -is already known to appear in the spin-3/2 and spin-5/2 cases on even-dimensional dS D for D ≥ 4 [18,53].
Note.Although unitarity is often considered to be equivalent to the positive-definiteness of the scalar product in the Hilbert space of mode solutions, this is not a sufficient requirement.For representation-theoretic unitarity, the scalar product must be both positive definite and invariant under the symmetry algebra (or group) of interest.
In this Section, the symmetries of interest correspond to the dS algebra, while, in Section 6, they correspond to the conformal-like so(4, 2) algebra.
Once we ensure both the unitarity and irreducibility of the so(4, 1) representations formed by the physical modes with fixed helicity, we will recall the so(4) content [Eqs.(4.11) and (4.12)] of these modes, as well as the value of the field-theoretic quadratic Casimir (2.12).
Then, it will be straightforward to identify the UIRs formed by the physical modes with a direct sum of Discrete Series UIRs of so(4, 1) [Eqs.(3.7) and (3.8)].

Infinitesimal dS transformations of physical modes and irreducibility of so(4, 1) representations
The infinitesimal dS transformations of the mode solutions can be studied with the use of the Lie-Lorentz derivative (2.9) with respect to the dS Killing vectors.Since the so(4) content of the so(4, 1) representations formed by mode solutions is already known (see Section 4), we just need to study the transformation properties of our mode solutions under dS boosts.In fact, it is sufficient to focus on just one dS boost (the reason is that the Lie bracket between a boost Killing vector and a rotational one is equal to another boost Killing vector).We choose to work with the following boost Killing vector: (t, θ 3 ) as linear combinations of other mode solutions 10 .There are (at least) two different ways we can follow in order to proceed: • i) Direct calculation, where in order to express L X Ψ (phys, ±ℓ; m;k) µ 1 ...µr as a linear combination of other modes, one has to use the raising and lowering differential operators for the angular momentum quantum number ℓ, as in Refs.[14,23,49,53].or • ii) Making use of the matrix elements of so(5) generators obtained by Gelfand and Tsetlin [43].More specifically, one can use these matrix elements to find explicit expressions for the so(5) transformations of tensor-spinor spherical harmonics on S 4 and then perform analytic continuation to dS 4 .
In this paper, we follow approach ii.Here we present the final expressions for L X Ψ (phys, ±ℓ; m;k) µ 1 ...µr .The reader who is familiar with so(D + 1) representations formed by tensor-spinor spherical harmonics on S D [55] can infer the results from Gelfand and Tsetlin's work [43].Technical details for the derivation can be found in Appendix A.
11 I would like to thank Atsushi Higuchi for pointing out that this current is conserved.
Gauge invariance of the scalar product.Let us show that, with respect to the scalar product (5.4),all pure gauge modes (4.29) are orthogonal to themselves, as well as to all physical modes.In particular, letting Ψ (2) µ 1 ...µr be a pure gauge mode (4.29) -i.e.Ψ (2) , where we have omitted the quantum number labels for convenience -the current (5.3) can be expressed as where Ψ (1) is any physical or pure gauge mode.As J µ (Ψ (1) , Ψ (pg) ) in Eq. (5.9) is equal to the divergence of an anti-symmetric tensor, the scalar product between any pure gauge mode and any other mode is always zero.Also, this directly implies that the scalar product (5.4) is invariant under restricted gauge transformations (2.17).
Thus, the set of all physical mode solutions for the strictly massless spin-(r + 1/2) ≥ 3/2 theory, satisfying Eqs.(2.15) and (2.16), corresponds to the direct sum of Discrete Series UIRs D − (r 12 .This is in agreement with the "field theoryrepresentation theory dictionary" suggested previously by us [18].

Conformal-like symmetries for strictly massless fermions
In this Section, we present our main results, i.e. we present and study new conformal-like symmetries for strictly massless spin-s ≥ 3/2 fermions on dS 4 .Conformal Killing vectors of dS 4 .For later convenience, let us review the basics concerning the conformal Killing vectors on dS 4 .The five conformal Killing vectors of dS 4 satisfy with ∇ α V α ̸ = 0. (The ten dS Killing vectors, ξ µ , satisfy the same equation, but they are divergence-free.)The 15-dimensional Lie algebra generated by the dS Killing vectors and the conformal Killing vectors is isomorphic to so(4, 2).The Lie bracket between a dS Killing vector and a conformal Killing vector is equal to a conformal Killing vector, while the Lie bracket between two conformal Killing vectors closes on so(4, 1).These facts can be understood from the so(4, 2) commutation relations: with A ′ , B ′ , C ′ , D ′ = −1, 0, ..., 4, where . The generators M −1A ′ , with A ′ = 0, ..., 4, can be identified with the five conformal Killing vectors of dS 4 , while the generators M A ′ B ′ , with A ′ , B ′ = 0, ..., 4, can be identified with the ten dS Killing vectors.
Each of the five conformal Killing vectors of dS 4 , denoted for convenience as V (0)µ , V (1)µ , ..., V (4)µ , can be expressed as a gradient of a scalar function13 Each of the five scalar functions ϕ V (A) (A = 0, 1, ..., 4) satisfies i.e. ∇ µ V (A) ν = −g µν ϕ V (A) .The scalar functions satisfying Eq. (6.4) can be found by analytically continuing the scalar functions that are annihilated by the operator ∇ µ ∇ ν + g µν on S 4 14 .If we embed dS 4 in 5-dimensional Minkowski space as −(X 0 ) 2 + 4 j=1 (X j ) 2 = 1, then the five scalar functions ϕ V (A) are ϕ V (A) = X A (this equality holds up to a proportionality constant, which we ignore in the present paper).In the case of global coordinates (4.1) we have Below we will often drop the label '(A)' from V (A)µ and ϕ V (A) .Thus, from now on, we will denote conformal Killing vectors of dS 4 as V µ = ∇ µ ϕ V or W µ = ∇ µ ϕ W , unless otherwise stated.
The differential operator T V maps solutions of Eqs.(2.15) and (2.16) into other solutions, i.e.T V corresponds to a symmetry of these equations.
Note.The term in the last line of Eq. (6.6) does not correspond to a restricted gauge transformation (2.17).This can be understood by observing that the gauge function, λ µ 2 ...µr , in the restricted gauge transformation (2.17) satisfies Eq. (2.18), while γ 5 Ψ µ 2 ...µrρ V ρ in the last line of Eq. (6.6) does not; it satisfies the following equation instead 15 : 14 It is known that such functions on S 4 exist [21].More specifically, they correspond to scalar spherical harmonics on S 4 [14]. 15Although the term in the second line of Eq. (6.6) is not a restricted gauge transformation, it still corresponds to an "off-shell" gauge transformation -by "off-shell" gauge transformation we mean any gauge transformation that leaves invariant the Lagrangian for strictly massless fermions (the restricted gauge transformations (2.17) correspond to a special case of the "off-shell" transformations).Hermitian and gauge-invariant Lagrangians for strictly massless fermions on AdS4 have been constructed in Ref. [32] (see also Ref. [26]).By analytically continuing AdS4 to dS4, i.e. by replacing the AdS radius as R AdS → iR dS , In order to prove that the conformal-like transformation (6.6) corresponds to a symmetry we need to show that T V Ψ µ 1 ...µr satisfies the same field equations as Ψ µ 1 ...µr , i.e.Eqs.(2.15) and (2.16).It is convenient to define the totally symmetric tensor-spinors ∆ V Ψ µ 1 ...µr and P V Ψ µ 1 ...µr as and such that We observe that ∆ V Ψ µ 1 ...µr and P V Ψ µ 1 ...µr have opposite gamma traces Thus, the gamma-tracelessness property of the conformal-like transformation (6.6), is straightforwardly shown.Now let us show that, if Ψ µ 1 ...µr satisfies Eq. (2.15), then so does T V Ψ µ 1 ...µr .In other words, we will show that T V Ψ µ 1 ...µr is an eigenfunction of the Dirac operator with eigenvalue −ir.Acting with the Dirac operator on ∆ V Ψ µ 1 ...µr and P V Ψ µ 1 ...µr , we find and respectively, where we have used Eq.(2.8).Adding Eqs.(6.11) and (6.12) by parts, and making use of Eqs.(6.9) and (6.10), we find where R dS is the dS radius (R dS = 1 in our units), one can extend the Lagrangians for strictly massless fermions on AdS4 [32] to gauge-invariant, but non-hermitian, Lagrangians on dS4.The field equations derived from these non-hermitian Lagrangians on dS4 are invariant under "off-shell" gauge transformations that have the form δΨµ 1 ...µr = (∇ (µ 1 + i 2 γ (µ 1 )χ µ 2 ...µr ) , where χµ 2 ...µr is a totally symmetric tensor-spinor with γ µ 2 χµ 2 ...µr = 0.If one specialises to the TT gauge, these field equations reduce to Eqs. (2.15) and (2.16), while the initial "off-shell" gauge invariance reduces to the restricted gauge invariance with gauge transformations given by (2.17).
6.2 Conformal-like so(4, 2) algebra generated by the dS symmetries and the conformal-like symmetries In order to understand the structure of the algebra generated by the dS transformations (2.9) and the conformal-like transformations (6.6) we need to study the corresponding Lie brackets (i.e.commutators).Below, V µ = ∇ µ ϕ V and W µ = ∇ µ ϕ W denote any two conformal Killing vectors of dS 4 [see Eq. ( 6. 3)].
Commutator between dS and conformal-like transformations.After a straightforward calculation, the commutator between a dS transformation (2.9) and a conformal-like transformation (6.6) is found to be where [ξ, V ] is the Lie bracket between the Killing vector ξ and the conformal Killing vector V , i.e. [ξ, V ] µ = L ξ V µ (L ξ is the usual Lie derivative with respect to ξ).
Commutator between two conformal-like transformations.The calculation of the commutator between two conformal-like transformations, [T W , T V ]Ψ µ 1 ...µr , is quite long.Thus, here we present the final result and we refer the reader to Appendix B for some details of the calculation.The result is where The second term on the right-hand side of Eq. (6.16) is a restricted gauge transformation of the form (2.17), where Structure of the conformal-like algebra.To conclude, the structure of the conformallike algebra generated by the ten dS transformations (2.9) and the five conformal-like transformations (6.6) is determined by the following commutation relations: where L µ 2 ...µr is given by (6.17), ξ µ and ξ ′µ are any two dS Killing vectors, while W µ = ∇ µ ϕ W and V µ = ∇ µ ϕ V are any two conformal Killing vectors.The commutation relations (6.18a)-(6.18c)coincide with the so(4, 2) commutation relations (6.2) up to the restricted gauge transformation in Eq. (6.18c).
Our results demonstrate that there is a representation of so(4, 2) (which closes up to field-dependent gauge transformations) acting on the solution space of Eqs.(2.15) and (2.16).In the following Subsection, we will show that the physical modes, which have been shown to form a direct sum of so(4, 1) UIRs (see Section 5), also form a direct sum of so(4, 2) UIRs.
• Note.One might think that the closure of the conformal-like algebra up to (fielddependent) gauge transformations is a consequence of the term in the second line of Eq. (6.6).In order to argue that this is not the case, let us focus on the strictly massless spin-3/2 field and depart from the TT gauge: Consider the full Rarita-Schwinger (RS) equation for the strictly massless spin-3/2 field (gravitino) on dS 4 [25] where . This equation is invariant under "off-shell" gauge transformations In other words, if ψ µ satisfies the RS equation, then so does ∆ V ψ µ .Because of the "off-shell" gauge symmetry (6.20), Eq. (6.21) does not include a part corresponding to the second line of Eq. (6.6).Then, the commutator between two conformal-like transformations (6.21) is found to be where we notice the appearance of an "off-shell" gauge transformation (which is not a restricted gauge transformation (2.17)) on the right-hand side.The rest of the structure of the symmetry algebra is determined by the same commutation relations as in Eqs.(6.18a) and (6.18b) (with T V replaced by ∆ V ).

Conclusion.
As in the TT gauge, the full RS equation (6.19) enjoys a conformal-like so(4, 2) symmetry and the algebra closes up to "off-shell" gauge transformations (6.20) that do not correspond to restricted gauge transformations (2.17).However, in the TT case (6.18c), the algebra closes up to restricted gauge transformations.
7 The physical modes also form UIRs of the conformal-like algebra In this Section, we show that the 'positive frequency' physical modes (4.15) and (4.24) of the strictly massless spin-s ≥ 3/2 fermionic theories form UIRs of the conformal-like so(4, 2) algebra.To be specific: • The irreducibility of the so(4, 2) representations will be demonstrated by showing that the physical modes with fixed helicity transform among themselves under the infinitesimal conformal-like transformations (6.6).In particular, the physical modes with helicity +s [Eq.(4.24)], and the ones with helicity −s [Eq.(4.15)], will be shown to separately form irreducible representations of so(4, 2).(Recall that we have already shown that these modes form a direct sum of UIRs of the dS algebra so(4, 1) -see Section 5.) • As for showing the unitarity of the two aforementioned irreducible so(4, 2) representations, we work as follows.First, we recall from Section 5 that the physical modes with helicity ∓s form a so(4, 1) UIR with dS invariant and positive definite scalar product given by (±1)×(5.4).Then, since a positive definite and so(4, 1)-invariant scalar product is known, it is sufficient to show that this scalar product is also invariant under the conformal-like symmetries (6.6).

Conformal-like transformations of physical modes and irreducibility of so(4, 2) representations
Let us start with the simple observation that, according to Eq. (6.2), the Lie bracket between a conformal Killing vector and a dS Killing vector is equal to a conformal Killing vector.Similarly, the commutator [L ξ , T V ]Ψ µ 1 ...µr in Eq. (6.18b) is equal to a conformallike symmetry transformation.Thus, as the so(4, 1) representation-theoretic properties of the physical modes are known (see Section 5), it is sufficient to study just one of the five conformal-like transformations (6.6) for our physical modes.Then, the transformation properties of the physical modes under the rest of the conformal-like transformations can be found using the commutation relations (6.18b).

so(4, 2)-invariant scalar product and unitarity
In the previous Subsection, we showed that the physical modes form a direct sum of irreducible representations of the conformal-like algebra.The only remaining step for showing that this is a direct sum of so(4, 2) UIRs is to ensure the existence of a so(4, 2)-invariant and positive definite scalar product.

Conformal-like transformations of field strength tensor-spinors
In order to gain some insight into the interpretation of the conformal-like transformations T V Ψ µ 1 ...µr (6.6), we study the corresponding transformations of the field strength tensorspinors (i.e.curvatures).In particular, we study the transformations of the spin-s = 3/2, 5/2 field strengths explicitly, while in the spin-s ≥ 7/2 cases we make a conjecture for the expressions of the transformations.

Spin-3/2 field strength tensor-spinor
The field strength tensor-spinor for the strictly massless spin-3/2 field is For later convenience, we will denote this as F µ 1 ν 1 (Ψ).The field strength F µ 1 ν 1 (Ψ) is invariant under not only restricted gauge transformations (2.17) but also "off-shell" gauge transformations (6.20).Useful properties.Let us discuss some of the properties of F µ 1 ν 1 (Ψ) that will be useful in studying its conformal-like transformation.Using the field equations (2.15) and (2.16) for Ψ ν , we find The dual field strength tensor-spinor is defined as Expressing ϵ κλ µ 1 ν 1 in Eq. (8.3) in terms of gamma matrices [see Eq. (2.5)], and using the gamma-tracelessness of F µ 1 ν 1 (Ψ), we find Also, a straightforward calculation shows that the following identity holds: It is easy to show that each of the two terms in this equation is zero by observing that 17 It immediately follows from Eqs. (8.4)- (8.6) that Conformal-like transformation.After a straightforward calculation, the conformal-like transformation of the field strength, F µ 1 ν 1 (T V Ψ), is expressed as where in the first line we have used T V Ψ µ = ∆ V Ψ µ +P V Ψ µ [see Eq. (6.9)] and F µ 1 ν 1 (P V Ψ) = 0 (the latter follows from the gauge-invariance of the field strength).Then, using Eq.(8.7), we find or equivalently where L V is the Lie-Lorentz derivative (2.9) with respect to the conformal Killing vector V (6.3) 18 .

Conclusion.
The expression (8.10) makes clear that the conformal-like transformation of the spin-3/2 field strength tensor-spinor corresponds to the product of two transformations: an infinitesimal axial rotation (i.e.multiplication with γ 5 ) times an infinitesimal conformal transformation (i.e.Lie-Lorentz derivative plus a conformal weight term).

Spin-5/2 field strength tensor-spinor
The field strength tensor-spinor for the strictly massless spin-5/2 field is a rank-4 tensorspinor given by (8.11) 17 Proof of Eq. (8.6).In order to prove Eq. (8.6), we contract and we use the definition (8.3) of the dual field strength.Then, using well-known identities for ϵ µ 1 ν 1 αβ , while also using the divergence-freedom of the field strength, we can show that ϵ σκ , we arrive at Eq. (8.6).End of proof. 18The infinitesimal Lorentz transformation term ∇αV β γ αβ /4 in the Lie-Lorentz derivative LV in Eq. (8.10) vanishes because, according to Eq. ( 6.3), ∇ [α V β] = 0.This is symmetric under the exchange of pairs of indices (8.12) It is also anti-symmetric in its first two and last two indices and satisfies the identity As in the spin-3/2 case, the field strength is invariant under not only restricted gauge transformations (2.17) but also gauge transformations of the following form: , where ϵ ν is an arbitrary vector-spinor.
Working as in the spin-3/2 case, we can show that the spin-5/2 field strength (8.11) is gamma-traceless and divergence-free with respect to all of its indices, and it also satisfies the identities Conformal-like transformation.Let us find the conformal-like transformation of the field strength, F µ 1 ν 1 µ 2 ν 2 (T V Ψ).The calculation is similar to the spin-3/2 case, but quite longer.The result is or equivalently Conclusion.As in the spin-3/2 case (8.10), the expression (8.18) makes clear that the conformal-like transformation of the spin-5/2 field strength corresponds to the product: infinitesimal axial rotation times infinitesimal conformal transformation.as the gauge-invariant rank-2r tensor-spinor that satisfies and it is also anti-symmetric under the exchange of the indices µ l ↔ ν l for l = 1, ..., r.It is also symmetric under the exchange of any two pairs of indices as in the following example: ...µ r−1 ν r−1 µ 1 ν 1 and so forth, (8.20) while it also satisfies the identities and Conjecture.The conformal-like transformation of the spin-(r + 1/2) ≥ 7/2 field strength tensor-spinor is given by or equivalently We also showed that the physical (positive frequency) mode solutions (4.15) and (4.24) form a direct sum of UIRs of the conformal-like so(4, 2) algebra.As for the interpretation of the conformal-like symmetries, we found that, at the level of the field strength tensor-spinors, each conformal-like transformation is expressed as a product of two transformations: an infinitesimal axial rotation and an infinitesimal conformal transformation (this was shown explicitly for the spin-s = 3/2, 5/2 cases and conjectured for the cases with s ≥ 7/2 -see Section 8).Let us discuss in passing the flat-space limit of the conformal-like symmetries (i.e. the limit of zero cosmological constant).First, we observe that the flat-space limit of the five conformal Killing vectors (6.3) of dS 4 gives rise to the four translation Killing vectors and the generator of dilations of Minkowski spacetime (rather than the five conformal Killing vectors of Minkowski spacetime as one might expect).This can be verified by recovering the dS radius, R dS , such that (4.1) is written as The five de Sitterian conformal-like symmetries (6.6) reduce to the following flat-space symmetries of Eq. ( 9.1) where w ρ is a translation Killing vector or the generator of dilations (i.e., in the standard Minkowski coordinates x 0 , x 1 , x 2 , x 3 with line element −(dx 0 ) 2 + 3 j=1 (dx j ) 2 , we have We observe that the transformation (9.2) is a product of two transformations.However, unlike in dS 4 , in Minkowski spacetime, each of the two transformations present in the product (9.2) is also a symmetry.In other words, Eqs.(9.1) are invariant under the replacement Ψ µ 1 ...µr → γ 5 Ψ µ 1 ...µr (infinitesimal axial rotations), as well as under Ψ µ 1 ...µr → w ρ ∂ ρ Ψ µ 1 ...µr .
In Ref. [47], using the unfolded formalism, Vasiliev presented a sp(8, R) invariant formulation of free massless fields (gauge potentials) of any spin in AdS 4 and showed that the free field equations are invariant under o(4, 2) (see also Ref. [48]).Although further study is required, it is likely that the dS version of Vasiliev's conformal invariance [47] is related to the conformal-like symmetries we presented in this paper.
It is worth recalling that unitary superconformal field theories on dS 4 are known to exist [19].In view of our newly discovered conformal-like symmetries for strictly massless fermions, it is interesting to look for new (and possibly unitary) supersymmetric theories on dS 4 that include strictly massless fermions of any spin s ≥ 3/2. 19nd from the WW Smith Fund.Last, but not least, I would like to thank Alex for reminding me that there exist poetic qualities in life beyond poems, which was at the very least inspiring, for lack of a better wor(l)d.
A Deriving Eq. (5.2) by analytically continuing so(5) rotation generators and their matrix elements to so(4, 1) The aim of this Appendix is to explain how to use group-theoretic tools and analytic continuation techniques in order to derive the transformation properties of physical modes in Eq. (5.2).
A.1 Background material for representations of so( 5) and Gelfand-Tsetlin patterns The representations of the algebra so(D + 1) -with arbitrary D -and the specification of the matrix elements of the generators have been studied by Gelfand and Tsetlin [43].
The D(D + 1)/2 generators I AB = −I BA (A, B = 1, 2, ..., D + 1) of so(D + 1) satisfy the commutation relations In Ref. [43], the action of the so(D+1) generators has been determined in the decomposition so(D + 1) ⊃ so(D).In particular, the representation space for a so(D + 1) representation is chosen to be the direct sum of the representation spaces of all representations of so(D) that appear in the so(D + 1) representation.(If a representation of so(D) appears in a representation of so(D +1), then it appears with multiplicity one.)Similarly, the generators of so(D) are determined in the decomposition so(D) ⊃ so(D − 1) and so forth.In other words, Gelfand and Tsetlin [43] determined a so(D +1) representation in the decomposition so(D + 1) ⊃ so(D) ⊃ ... ⊃ so(2).Focusing on so (5).We now specialise to so(5) -since this is the non-compact partner of the dS algebra so(4, 1).Let us review some basic results obtained by Gelfand and Tsetlin [43] (with slightly modified notation).A (unitary) irreducible representation of so( 5) is specified by the highest weight ⃗ s = (s 1 , s 2 ) with s 1 ≥ s 2 ≥ 0, where the numbers s 1 and s 2 are simultaneously integers or half-odd-integers.The 10 anti-hermitian generators I AB = −I BA (A, B = 1, ..., 5) act on a finite-dimensional vector space corresponding to a direct sum of so(4) representation spaces (as described at the beginning of the Subsection).Let v denote the orthonormal basis vectors in the so(5) representation space.Each basis vector is uniquely labelled by a "Gelfand-Tsetlin pattern", α, as follows: The labels s 1 , s 2 are the same for all basis vectors, since they correspond to the highest weight specifying the so(5) representation.The rest of the labels in Eq. (A.2) specify the content of the so(5) representation concerning the chain of subalgebras so(4) ⊃ so(3) ⊃ so(2).In particular, the labels f 1 , f 2 correspond to a so(4) highest weight ⃗ f ≡ (f 1 , f 2 ) with f 1 ≥ |f 2 |, where f 1 and f 2 are both integers or half-odd integers, while f 2 can be negative.The so(3) weight p ≥ 0 is an integer or half-odd integer.The full basis of the representation space is given by all v(α)'s in eq.(A.2) -with fixed s 1 , s 2 -satisfying: 3) The numbers s 1 , s 2 , f 1 , f 2 , p and q are all integers or half-odd integers.
In order to obtain the desired transformation formulae (5.2) using analytic continuation, we need to study the action of the generator I 54 on the basis vectors (A.2).This is given by [43]: (Our matrix elements differ from the matrix elements of Ref. [43] by a factor of 1/2.)Note A.2 Specialising to so(5) representations formed by tensor-spinor spherical harmonics on S 4 The line element of S 4 can be parametrised as where 0 ≤ θ 4 ≤ π and dΩ 2 is the line element of S 3 (4.2).For later convenience, note that the line element (A.7) can be analytically continued to the dS 4 line element (4.1) by making the replacement -the variable x has been already introduced in Eq. (4.18).Let / ∇ = γ µ ∇ µ be the Dirac operator on S 4 , where γ µ and ∇ µ are the gamma matrices and covariant derivative, respectively, on S 4 .We are interested in (totally symmetric) rank-r tensor-spinor spherical harmonics ψ(n; r, σℓ; m;k) (A.12) The numbers ℓ, m and k are the angular momentum quantum numbers on S 3 , S 2 and S 1 , respectively, and their allowed values are found from (A.3).
Based on the discussion in the previous paragraph, we can identify each eigenmode ψ(n; r, σℓ; m;k) µ 1 ...µr (θ 4 , θ 3 ) with a basis vector (A.2) labeled by the pattern (A.12).In particular, we make the identifications: where the phase factor −i(−1) r has been introduced for convenience.
A.3 Transformation properties of tensor-spinor spherical harmonics on S 4 under so(5) In this Subsection, we find the so(5) transformation formulae for L S ψ(n; r=r, ±ℓ; m;k) µ 1 ...µr that (after analytic continuation) will give rise to the so(4, 1) transformation formulae (5..14).However, these transformation properties refer to normalised eigenmodes, while the desired dS transformation properties (5.2) refer to un-normalised eigenmodes.Therefore, we will first find the so(5) transformation properties for the un-normalised eigenmodes on S 4 (the un-normalised eigenmodes will be defined below), and then perform analytic continuation to dS 4 .
B Details for the computation of the commutator (6.16)