Bulk interactions and boundary dual of higher-spin-charged particles
aa r X i v : . [ h e p - t h ] S e p Bulk interactions and boundary dual of higher-spin-chargedparticles
Adrian David ∗ and Yasha Neiman † Okinawa Institute of Science and Technology,1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan (Dated: September 8, 2020)
Abstract
We consider higher-spin gravity in (Euclidean)
AdS , dual to a free vector model on the 3dboundary. In the bulk theory, we study the linearized version of the Didenko-Vasiliev black holesolution: a particle that couples to the gauge fields of all spins through a BPS-like pattern ofcharges. We study the interaction between two such particles at leading order. The sum overspins cancels the UV divergences that occur when the two particles are brought close together,for (almost) any value of the relative velocity. This is a higher-spin enhancement of supergravity’sfamous feature, the cancellation of the electric and gravitational forces between two BPS particlesat rest. In the holographic context, we point out that these “Didenko-Vasiliev particles” are justthe bulk duals of bilocal operators in the boundary theory. For this identification, we use thePenrose transform between bulk fields and twistor functions, together with its holographic dualthat relates twistor functions to boundary sources. In the resulting picture, the interaction betweentwo Didenko-Vasiliev particles is just a geodesic Witten diagram that calculates the correlator oftwo boundary bilocals. We speculate on implications for a possible reformulation of the bulk theory,and for its non-locality issues. ∗ Electronic address: [email protected] † Electronic address: [email protected] ontents I. Introduction and summary II. Free HS fields and local boundary sources
EAdS III. HS fields with bulk particle sources R EAdS IV. HS interaction between two bulk particles R case 24C. The EAdS case 27 V. Twistors, HS algebra and boundary correlators
VI. The Didenko-Vasiliev solution and boundary bilocals
VII. Discussion Acknowledgements References . INTRODUCTION AND SUMMARY
Higher-spin (HS) gravity [1, 2] is a smaller sibling of string theory. It is a theory ofinfinitely many massless fields with different spins, including a spin-2 “graviton”. Whereasstring theory is holographically dual to matrix-like conformal field theories [3–5], higher-spin gravity is dual to vector models [6–8]. While HS gravity can be defined in variousdimensions, our interest will be the 4d case. Specifically, we will consider the type-A theory,which has one field of every integer spin, and is holographically dual to a free vector modelof N complex scalar fields [6].From a complementary point of view, HS gravity is a larger sibling of supergravity, thelow-energy limit of string theory. Whereas supergravity extends the spacetime symmetryof GR in a fermionic direction, HS gravity extends it in a bosonic direction, resulting in aninfinite multiplet of massless “partners” for the graviton with different spins. In supergravitywith N = 2 or higher supersymmetry, particular importance is placed on extremal blackhole (or black brane) solutions [9] that saturate the BPS bound. Such black holes definebackgrounds that preserve part of the theory’s supersymmetry. The gravitational chargeof these black holes, i.e. their mass, is proportional to their electric charge under gravity’sspin-1 superpartner. These supergravity solutions play a key role in string theory, wherethey are ultimately identified [10] with D-branes [11, 12].In HS gravity, a similar object is known – the Didenko-Vasiliev “black hole” [13] (see alsogeneralizations in [14]). This is a spherically symmetric solution to the Vasiliev equations,constructed by analogy with the Kerr-Schild procedure for the Schwarzschild black hole.Aside from this formal similarity, it’s not at all clear that this solution behaves like thefamiliar black holes of GR, with their event horizons and thermodynamical properties; hencethe quotation marks around the term “black hole”. The Didenko-Vasiliev black hole ischarged under the HS fields of all spins. These charges are all proportional to each other,reflecting a partial preservation of HS symmetry, in clear analogy with the supergravity case.Though we won’t consider it here, there is also a supersymmetric version of the Didenko-Vasiliev solution, where the type-A and type-B HS gravities are naturally combined intoan N = 2 supermultiplet. One then finds [13] that the solution preserves a quarter of thesupersymmetries, further strengthening the analogy to BPS black holes.In this paper, we will study the linearized version of the Didenko-Vasiliev black hole,3hich was described by the same authors in [15]. This is a solution to Fronsdal’s equationsfor free HS gauge fields [16, 17], with a source localized at the spatial origin (or, from aspacetime perspective, along the time axis). We will refer to this source as a “Didenko-Vasiliev particle”, or DV particle for short. Our main case of interest, and the originalcontext of [15], will be DV particles in AdS , where a non-linear HS theory exists, completewith holographic duals. However, we will also consider DV particles in flat spacetime, wheresome calculations simplify. We will work in Euclidean signature, so that our spacetime willbe either flat R or hyperbolic space, i.e. Euclidean AdS ( EAdS for short).The main object of our calculations will be the leading-order action from two DV particlesinteracting via their HS gauge fields. For every spin separately, this action is IR-divergentin R , but finite in EAdS . When the particles are brought close together, a UV divergencearises; this is the same as the IR divergence from the flat case, viewed from a differentperspective. However, we find that these divergences cancel upon summing over spins,thanks to the DV particles’ special pattern of charges. This is an enhanced version of thewell-known cancellation of the electric and gravitational forces between two BPS objects insupergravity (or between two extremal Riessner-Nordstrom black holes). In the latter case,the cancellation only holds when the two objects are mutually at rest; the introduction of arelative velocity reveals the different tensor structure of the two forces, and they no longercancel. In our case, the cancelation holds for any relative velocity, with the single exceptionof a particle and antiparticle mutually at rest. This criterion can in fact be used to derive the DV pattern of charges (we are grateful to Slava Lysov for calling our attention to thisfeature). Either way, it reflects a certain kind of non-locality, or softness, of interactionsgoverned by HS symmetry.Our next observation is that the higher-spin fields of the DV particle, as given in [15],play a role in higher-spin holography. Since the original motivation in [13, 15] was to mimicblack hole solutions of GR, one might expect that the relevant holographic duality would bebetween non-perturbative black holes in the bulk and thermal states in the boundary CFT.We make no claims here about the validity of this scenario. Instead, we point out a differentone, in which the linearized DV solution appears in a perturbative role. Specifically, theDV particle describes the linearized bulk solution, i.e. the “boundary-to-bulk propagator”,that corresponds to a bilocal operator [18, 19] in the boundary theory. In this picture, theworldline of the DV particle, which sources its HS fields, lies on the bulk geodesic that4onnects the two boundary “legs” of the bilocal – see figure 1. Note that this can’t directlyapply in the original setup of [13, 15]: there, the DV particle’s worldline is a timelike geodesicin
AdS , which has no boundary endpoints. Thus, we should either work in de Sitter space,or consider spacelike geodesics. In this paper, we will work for simplicity in Euclidean EAdS ; there, all geodesics are spacelike, and have boundary endpoints.Our identification of the DV particle with a boundary bilocal operator is obtained via thespacetime-independent twistor formalism for HS holography, developed in [20, 21]. Thoughthe appearance of a particle-like bulk source in the context of a “boundary-to-bulk propa-gator” may sound surprising, it is in fact a special case of the recently uncovered relation[22–24] between OPE blocks in the boundary CFT and geodesic Witten diagrams. In par-ticular, the interaction of two bulk DV particles in EAdS , which we will calculate here, isnothing but a geodesic Witten diagram for the correlator of two boundary bilocals.The rest of the paper is structured as follows. In section II, we review Fronsdal’s theoryof free HS fields, in flat space and EAdS , including boundary-to-bulk propagators and2-point functions. In section III, we construct and solve the field equations for linearizedHS fields sourced by a bulk particle that travels along a geodesic worldline. In section IV,we study the action for two such particles interacting via HS fields, and demonstrate thecancellation of UV divergences for the DV pattern of charges. In the flat case, we find theaction analytically; in EAdS , we end up with an unpleasant integral, which however agreesnumerically with a simple analytic answer guessed from holography.In section V, we take a detour to introduce EAdS twistors, higher-spin algebra, thePenrose transform, as well as HS-algebraic formulas for boundary-to-bulk propagators andboundary n -point functions. In this, we will follow the formalism of [20, 21, 25], whichcombines HS algebra, embedding space, and spacetime-independent twistors. With thismachinery in place, we proceed to make our main claims in section VI. There, we evaluatethe linearized bulk fields that correspond to a bilocal operator in the boundary CFT, andnotice that they coincide with the fields of a DV particle. We further notice that theinteraction of two such particles, as calculated in section IV, reproduces the correlator oftwo boundary bilocals, in the spirit of the general theory of geodesic Witten diagrams [22].Finally, in section VII, we speculate about the relevance of our construction to understandinginteracting HS theory and its locality issues.5 I. FREE HS FIELDS AND LOCAL BOUNDARY SOURCES
In this section, we review the theory of free HS gauge fields, before introducing HS-chargedparticles in section III.
A. Fronsdal action, field equations and gauge symmetry
The theory of linearized higher-spin gauge fields on maximally symmetric spacetimes,with or without cosmological constant, was put forward by Fronsdal in [16, 17]. It generalizesMaxwell theory (which constitutes the spin-1 case) and linearized GR (the spin-2 case). Alsoincluded is the spin-0 case of a conformally massless scalar, though it is not strictly speakinga gauge theory.We will work in 4d Euclidean spacetime with a positive-definite metric g µν . The specificspacetime will be either flat R , or Euclidean AdS of unit radius. The commutator ofcovariant derivatives ∇ µ in these spacetimes reads:[ ∇ µ , ∇ ν ] v ρ = R v [ µ δ ρν ] EAdS . (1)A spin- s gauge potential in Fronsdal’s formulation is given by a tensor h µ ...µ s that is totallysymmetric and double-traceless in its indices: h µ ...µ s = h ( µ ...µ s ) ; h νρνρµ ...µ s − = 0 . (2)The first of these constraints becomes non-trivial for s ≥
2, and the second – for s ≥
4. Thelow-spin cases h , h µ and h µν correspond respectively to a scalar field, a Maxwell potentialand a linearized metric perturbation. For s ≥
1, the fields h µ ...µ s are subject to a gaugesymmetry: h µ ...µ s → h µ ...µ s + ∇ ( µ θ µ ...µ s ) , (3)where the gauge parameter θ µ ...µ s − is in turn constrained to be totally symmetric andtraceless (a constraint that becomes non-trivial for s ≥ θ µ ...µ s − = θ ( µ ...µ s − ) ; θ ννµ ...µ s − = 0 . (4)6ne can construct from h µ ...µ s a gauge-invariant second-derivative object, known as theFronsdal tensor. In flat space, this reads: F µ ...µ s = (cid:3) h µ ...µ s − s ∇ ( µ ∇ ν h µ ...µ s ) ν + s ( s − ∇ ( µ ∇ µ h νµ ...µ s ) ν , (5)where (cid:3) = ∇ µ ∇ µ . In EAdS , we get additional terms due to the curvature: F µ ...µ s = ( (cid:3) + 2 − s ) h µ ...µ s − s ∇ ν ∇ ( µ h µ ...µ s ) ν + s ( s − ∇ ( µ ∇ µ h νµ ...µ s ) ν = ( (cid:3) + 2 + 2 s − s ) h µ ...µ s − s ∇ ( µ ∇ ν h µ ...µ s ) ν + s ( s − ∇ ( µ ∇ µ h νµ ...µ s ) ν − s ( s − g ( µ µ h νµ ...µ s ) ν , (6)where the difference between the two expressions is in the ordering of the derivatives in thesecond term. For s = 1, F µ is the divergence of the Maxwell field strength; for s = 2, F µν isproportional to the linearized Ricci tensor. The free field equations for all spins are simply F µ ...µ s = 0.The linear equations of motion F µ ...µ s = 0 can be derived from a quadratic Lagrangianof the form ∼ h µ ...µ s F µ ...µ s . However, for s ≥
2, this Lagrangian is not invariant under thegauge transformation (3). We must instead use a trace-modified version of F µ ...µ s : G µ ...µ s = F µ ...µ s − s ( s − g ( µ µ F νµ ...µ s ) ν ; F µ ...µ s = G µ ...µ s − s g ( µ µ s G νµ ...µ s ) ν , (7)which satisfies a (partial) conservation law: ∇ ν G νµ ...µ s − = trace terms only . (8)For s = 2, this is just the construction of the Einstein tensor out of the Ricci tensor. For spins1 and 2, G µ ...µ s is conserved as usual. For s ≥
3, it is not the full divergence ∇ ν G νµ ...µ s − that vanishes, but rather its totally traceless part . This is sufficient to define an action: S = − Z d x √ g h µ ...µ s G µ ...µ s + boundary terms , (9)which is invariant (up to boundary terms) under the gauge transformation (3) subject tothe constraint (4) on the gauge parameter. The Euler-Lagrange equations of motion arenow G µ ...µ s = 0, which is of course equivalent to F µ ...µ s = 0. For more detailed discussionof free HS Lagrangians, see e.g. [26, 27]. 7or solutions to the source-free field equations F µ ...µ s = 0, it’s possible to choose atransverse traceless gauge. The gauge conditions and field equations can then be summarizedas: h ννµ ...µ s − = 0 ; ∇ ν h νµ ...µ s − = 0 ; (cid:3) h µ ...µ s = m h µ ...µ s , (10)where m = 0 in flat space, and m = s − s − EAdS .The gauge-invariant content of a solution to the field equations is captured by anotherinvariant tensor, this time involving s derivatives: ϕ µ ν ...µ s ν s = 2 s ∇ µ . . . ∇ µ s h ν ...ν s antisymmetrized over every µ k ν k pair,with all traces subtracted. (11)For s = 0, this is just the scalar field again, ϕ = h ; for s = 1, ϕ µν is the Maxwell fieldstrength; for s = 2, ϕ µνρσ is proportional to the linearized Weyl tensor. The field strength(11) is the sum of two chiral parts: one that is right-handed (i.e. self-dual) in every µ k ν k index pair, and one that is left-handed (i.e. anti-delf-dual). From the point of view of thefield strength (11), without referring to the gauge potential h µ ...µ s , the free massless fieldequations read: s = 0 : (cid:3) ϕ = m ϕ ; s = 1 : ∇ µ ϕ µν = ∇ [ µ ϕ νρ ] = 0 ; s ≥ ∇ µ ϕ µ ν ··· µ s ν s = 0 , (12)where the only difference between R and EAdS is now in the spin-0 case, with m = 0 , − B. Boundary data and on-shell action in
EAdS We now specialize to
EAdS spacetime. In this subsection, we will represent EAdS in Poincare coordinates ( z, x ), while writing the components of tensors in an orthonormalbasis. This is described by the vielbein: e z = e = e = e = 1 z . (13)All tensor indices will refer to the orthonormal basis. Greek indices ( µ, ν, . . . ) will takevalues in (0 , , , i, j, . . . ) take values in (1 , , EAdS , here in a flat conformal frame, is at z = 0. Theboundary behavior of solutions to the Fronsdal equations has beed studied e.g. in [28–30].The results are simplest in transverse traceless gauge, where the equations take the form(10). As usual, locally near the boundary, there are two branches of linearly independentsolutions, characterized by different powers of z , which are canonically conjugate to eachother. Global regularity on EAdS picks out a particular linear combination of the twobranches. We will refer to the branches as “electric” and “magnetic”, for reasons that willbecome clear. Their asymptotics, at leading order in small z , is defined by:Magnetic branch : h i ...i s ( z, x ) = z − s A i ...i s ( x ) ; (14)Electric branch : h i ...i s ( z, x ) = z s +1 J i ...i s ( x ) . (15)The other components of h µ ...µ s within each of the branches, i.e. the components whereone or more indices take the value 0, scale with higher powers of z , and are determinedby the components h i ...i s above. The tracelessness of h µ ...µ s then implies that A i ...i s and J i ...i s must be traceless. In addition, the electric boundary data J i ...i s is divergence-free, ∂ i J i i ...i s = 0. In the holographic duality with a free vector model of scalar fields, theelectric boundary data J i ...i s ( x ) describes the VEVs of the boundary theory’s single-traceoperators (i.e. spin- s conserved currents), while the magnetic data A i ...i s ( x ) describes thesources for these operators (i.e. spin- s gauge fields).Outside of transverse traceless gauge, it is helpful to characterize the two branches ofsolutions in a gauge-invariant way. For this purpose, we consider not the gauge potential h µ ...µ s , but its field strength, the generalized Weyl tensor ϕ µ ν ...µ k ν k from eq. (11). At everypoint, its linearly independent components are captured by a pair of totally symmetrictraceless tensors E i ...i s and B i ...i s , which describe respectively the field strength’s electricand magnetic parts: E i i ...i s = ϕ i i ... i s ; B i i ...i s = 12 ǫ i jk ϕ jk i ... i s . (16)On-shell, both of these are divergence-free in the 3d sense, i.e. ∂ i E i i ...i s = ∂ i B i i ...i s = 0.In the boundary limit z →
0, both E i ...i s and B i ...i s scale as z s +1 : E i ...i s ( z, x ) = z s +1 E i ...i s ( x ) ; B i ...i s ( z, x ) = z s +1 B i ...i s ( x ) . (17)The magnetic branch of solutions (14) can now be characterized by vanishing electric bound-ary data E i ...i s ( x ) = 0, while the electric branch (15) is characterized by vanishing magnetic9oundary data B i ...i s ( x ) = 0. Perhaps the quickest way to see this is to notice the behaviorof the different types of boundary data under the antipodal map z → − z : A i ...i s ( x ) and B i ...i s ( x ) are associated with antipodally even solutions, while J i ...i s ( x ) and E i ...i s ( x ) areassociated with antipodally odd ones [31, 32] (in fact, J i ...i s ( x ) and E i ...i s ( x ) are the same upto a numerical factor, while B i ...i s ( x ) is a conformal field strength for the 3d gauge potential A i ...i s ( x )).Returning now to transverse traceless gauge, let us work out the on-shell action of a freeHS field in EAdS . A detailed analysis can be found in [29]; here, we will take some shortcutstowards the final answer. Since we are dealing with free field theory, and since A i ...i s ( x ) and J i ...i s ( x ) are canonically conjugate, the action (with divergent pieces removed) should beproportional to R A i ...i s ( x ) J i ...i s ( x ) d x . As we will now show, the correct proportionalityfactor is: S [ h, h ] = 1 − s Z A i ...i s ( x ) J i ...i s ( x ) d x , (18)where the notation on the LHS is intended to emphasize that S is a quadratic form in thespace of free-field solutions h µ ...µ s ( z, x ). The numerical coefficient in (18) depends on ourchoice of boundary terms for the action, which in turn depend on our choice of boundaryconditions in the variational principle. Here, we are interested in the standard variationalprinciple for AdS/CFT, in which the source-type boundary data A i ...i s ( x ) is held fixed. Thevariation of the action (18) then reads: δS = (1 − s ) Z J i ...i s ( x ) δA i ...i s ( x ) d x . (19)Identifying this as a symplectic potential, we take another variation to extract the symplecticform: Ω = (1 − s ) Z δJ i ...i s ( x ) ∧ δA i ...i s ( x ) d x . (20)Note that in eq. (18), A i ...i s ( x ) and J i ...i s ( x ) were linearly related by the requirement ofregularity in EAdS , while in (20), we treat them as linearly independent.We can now justify the numerical coefficient in (18) by comparing the symplectic form (20)with the one derived directly from the Lagrangian − h µ ...µ s G µ ...µ s . In general, this will besomewhat complicated, due to the large number of terms with different index arrangementsinside G µ ...µ s . However, in the boundary limit z → G µ ...µ s beyond the “trivial” one (cid:3) h µ ...µ s can beignored. We then get the symplectic form:Ω = 1 z Z δh i ...i s ( z, x ) ∧ ∂ z δh i ...i s ( z, x ) d x , (21)where the integral is at a fixed small value of z . By linear superposition of (14) and (15),the general boundary behavior of h i ...i s is given by: h i ...i s ( z, x ) = z − s A i ...i s ( x ) + z s +1 J i ...i s ( x ) + . . . , (22)where the dots signify terms with A i ...i s ( x ) and its ∂ i derivatives multiplied by powers of z higher than z − s , and terms with J i ...i s ( x ) and its ∂ i derivatives multiplied by powers of z higher than z s +1 . In the symplectic form (21), only the terms explicitly written in (22)will contribute. The other terms will end up vanishing, either due to high powers of z , ordue to the wedge product’s antisymmetry along with integration over the boundary. All inall, we see that the symplectic form (21) ends up conciding with (20), with the numericalcoefficient arising as: 1 − s = 1 z (cid:0) z s +1 ∂ z z − s − z − s ∂ z z s +1 (cid:1) . (23) C. Embedding space, boundary-to-bulk propagators and 2-point function
While Poincare coordinates in
EAdS are sometimes useful, we prefer the more covariantembedding-space picture. There, EAdS is defined as the hyperboloid of unit timelike radiusembedded in a flat Minkowski space R , : EAdS = (cid:8) x µ ∈ R , | x µ x µ = − , x > (cid:9) . (24)The flat metric η µν of R , has mostly-plus signature. In an abuse of notation, we willdenote the 5d embedding-space indices by the same Greek letters ( µ, ν, . . . ) that we usedfor intrinsic tensors in the 4d spacetime. This is quite natural, because intrinsic EAdS vectors at a point x µ ∈ EAdS are simply vectors v µ ∈ R , that happen to be tangent tothe hyperboloid (24), i.e. that satisfy x · v ≡ x µ v µ = 0. In particular, the EAdS metricis simply g µν ( x ) = η µν + x µ x ν . Similarly, the EAdS covariant derivative ∇ µ is just theflat R , derivative ∂ µ , followed by projecting all tensor indices back into the hyperboloid,11sing the projector δ νµ + x µ x ν . With this notation, the formulas of section II A carry throughseamlessly.The conformal boundary of EAdS is defined by the projective lightcone in the R , embedding space, i.e. by null vectors ℓ µ ∈ R , , ℓ · ℓ = 0, modulo equivalence underrescalings ℓ µ → ρℓ µ . The bulk → boundary limit can be described as x µ → ℓ µ /z , wherethe parameter z goes to zero (as the coincident notation suggests, the Poincare coordinate z near the boundary plays this role). Vectors on the 3d conformal boundary are describedby R , vectors v µ that are tangential to the lightcone, v · ℓ = 0, subject to the equivalencerelation v µ ∼ = v µ + αℓ µ . Since our axes in R , are orthonormal, many of the formulas fromsection II B carry over to this picture, with embedding-space indices ( µ, ν, . . . ) in place ofthe orthonormal boundary indices ( i, j, . . . ).Let us now use the embedding-space language to write down massless spin- s boundary-to-bulk propagators. These are solutions to the free bulk field equations that are characterizedby a “source” point ℓ µ on the boundary, and a null polarization vector λ µ at that point, sothat ℓ · ℓ = ℓ · λ = λ · λ = 0. The equivalence relation λ µ ∼ = λ µ + αℓ µ can be made manifestby replacing λ µ with the totally-null bivector M µν ≡ ℓ [ µ λ ν ] , which satisfies: M µν ℓ ν = M µν M νρ = M [ µν M ρ ] σ = 0 . (25)At a bulk point x µ , the boundary inputs ( ℓ µ , M µν ) induce three quantities: a scalar ℓ · x andtwo vectors ℓ µ ⊥ ≡ ℓ µ + ( ℓ · x ) x µ and m µ ≡ M µν x ν , which satisfy: m µ ℓ µ ⊥ = m µ m µ = 0 ; ℓ ⊥ µ = ∇ µ ( ℓ · x ) . (26)The “ ⊥ ” label on ℓ µ ⊥ indicates projection in perpendicular to x µ , i.e. into the EAdS tangentspace at x . The projection of M µν into the tangent space at x reads: M µν ⊥ ≡ M µν + 2 m [ µ x ν ] = 2 m [ µ ℓ ν ] ⊥ ℓ · x . (27)The spin- s boundary-to-bulk propagator is now given by: h µ ...µ s ( x ) = M µ ν x ν . . . M µ s ν s x ν s ( ℓ · x ) s +1 = m µ . . . m µ s ( ℓ · x ) s +1 . (28)It’s easy to verify that this satisfies Fronsdal’s equations (10) in transverse traceless gauge.The field strength (11) associated with this solution can be calculated by repeatedly applyingthe identities: m [ µ ∇ ν ] ( ℓ · x ) = 12 ( ℓ · x ) M ⊥ µν ; ∇ µ m ν = − M ⊥ µν ; m [ µ ∇ ν ] m ρ = 12 M ⊥ µν m ρ , (29)12nd noting that ∇ µ M ⊥ νρ = − g µ [ ν m ρ ] can be discarded as a trace piece. The result reads: ϕ µ ν ...µ s ν s ( x ) = (2 s − s − · M ⊥ µ ν . . . M ⊥ µ s ν s ( ℓ · x ) s +1 − traces . (30)The subtraction of traces is equivalent to leaving just the purely right-handed and purelyleft-handed parts of M ⊥ µ ν . . . M ⊥ µ s ν s . Thus, if we define the left/right-handed parts of M ⊥ µν as: M L/Rµν ≡ (cid:18) M ⊥ µν ± ǫ µν λρσ x λ M ρσ (cid:19) , (31)then the field strength (30) can be written as: s = 0 : ϕ ( x ) = 1 ℓ · x ; (32) s ≥ ϕ µ ν ...µ s ν s ( x ) = (2 s − s − · M Lµ ν . . . M Lµ s ν s + M Rµ ν . . . M Rµ s ν s ( ℓ · x ) s +1 , (33)where we included the spin-0 case separately. Note that for s = 0, the factorials in (30),(33)become ill-defined. However, we can analytically continue to continuous values of s (where-upon the factorials become Gamma functions), and then take the limit s →
0. We then seethat (30) (but not (33)) correctly reproduces the spin-0 case ϕ ( x ) = h ( x ) = 1 / ( ℓ · x ).Let’s now identify the asymptotic behavior of the boundary-to-bulk propagator (28). Ata boundary point ˆ ℓ = ℓ , the boundary data is purely electric, and can be read off directlyfrom eq. (28) as: J µ ...µ s (ˆ ℓ ) = M µ ν ˆ ℓ ν . . . M µ s ν s ˆ ℓ ν s ( ℓ · ˆ ℓ ) s +1 . (34)The magnetic boundary data for the propagator (28) takes the form of a delta functionsupported at ˆ ℓ = ℓ : s = 0 : A (ˆ ℓ ) = 4 π δ (ˆ ℓ, ℓ ) ; (35) s ≥ A µ ...µ s (ˆ ℓ ) = − π (2 s − s s !( s − δ (ˆ ℓ, ℓ ) λ µ . . . λ µ s . (36)The numerical coefficient in (36) is rather non-trivial to derive, and we won’t reproduce thederivation here. It’s been worked out, with small mistakes, in [28], as well as by one of thepresent authors in [32]. A correct derivation can now be found in the updated version of[32]. As with eq. (30) above, though the coefficient in (36) is ill-defined for s = 0, we canreproduce the spin-0 case (35) by making s continuous and then taking the limit s → ℓ µ , M µν = 2 ℓ [ µ λ ν ]1 ) and ( ℓ µ , M µν ), then their contribution to thequadratic action (18) reads: S [ h , h ] = 12 Z d ˆ ℓ A (ˆ ℓ ) J (ˆ ℓ ) = 2 π J ( ℓ ) = 2 π ℓ · ℓ (37)for s = 0, and: S [ h , h ] = 1 − s Z d ˆ ℓ A µ ...µ s (ˆ ℓ ) J µ ...µ s (ˆ ℓ ) = 2 π (2 s − s s !( s − λ µ . . . λ µ s J µ ...µ s ( ℓ )= ( − s π (2 s )!4 s ( s !) · ( M µν M µν ) s ( ℓ · ℓ ) s +1 (38)for s ≥
1. Again, (37) can be regarded a special case of (38), by analytically continuing tocontinuous s and then sending s → III. HS FIELDS WITH BULK PARTICLE SOURCES
In this section, we couple Fronsdal’s linearized HS fields in R and EAdS to a particle-likesource, with support on a geodesic worldline γ . A. Action and field equations
First, consider coupling the spin- s Fronsdal field to a general HS current T µ ...µ s : S = Z d x √ g (cid:18) − h µ ...µ s G µ ...µ s + h µ ...µ s T µ ...µ s (cid:19) + boundary terms . (39)The current T µ ...µ s inherits the algebraic properties of h µ ...µ s , i.e. we take it to be totallysymmetric and double-traceless. In addition, invariance under the gauge symmetry (3)demands that T µ ...µ s be conserved in the same sense as G µ ...µ s , i.e. that the traceless part of ∇ ν T νµ ...µ s − should vanish: ∇ ν T νµ ...µ s − = trace terms only . (40)Varying the action (39) with respect to h µ ...µ s , we obtain the field equations G µ ...µ s = T µ ...µ s . Rearranging the trace as in (7), we express these equations in terms of the Fronsdal14ensor: F µ ...µ s = T µ ...µ s − s g ( µ µ s T µ ...µ s ) νν . (41)Now, what HS current T µ ...µ s can we associate with a point particle? Our first building blockis the delta function R γ dτ δ ( x, x ′ ) that localizes the particle on its worldline γ ; here, x µ arethe worldline’s coordinates, and dτ = p dx µ dx µ is the length element. The second buldingblock is the particle’s 4-velocity u µ = dx µ /dτ . We will assume minimal coupling, whichforbids spacetime derivatives of the delta function. We further assume that the worldlineis a geodesic, so that any further τ derivatives of u µ vanish. The most general totallysymmetric tensor then reads: T µ ...µ s ( x ′ ) = Z γ dτ δ ( x, x ′ ) ⌊ s/ ⌋ X n =0 Q ( s ) n g ( µ µ . . . g µ n − µ n u µ n +1 . . . u µ s ) , (42)with some coefficients Q ( s ) n . The n = 0 term in (42) is conserved, thanks to the geodesiccondition u ν ∇ ν u µ = 0. The n ≥ T µ ...µ s is double-traceless overall. Their divergence is automatically a tracepiece, so they do not affect the conservation condition (40). This leaves the n = 1 term,which does not satisfy the conservation law (40), and must therefore be ruled out. All in all,we end up with the simplest possible coupling between the spin- s field and a point particle: T µ ...µ s ( x ′ ) = Q ( s ) Z γ dτ δ ( x, x ′ ) u µ . . . u µ s − double traces ; (43) S = − Z d x √ g h µ ...µ s G µ ...µ s + Q ( s ) Z γ dτ h µ ...µ s u µ . . . u µ s + boundary terms , (44)where we renamed Q ( s )0 ≡ Q ( s ) for brevity. The field equations (41) in the presence of thecurrent (43) read: F µ ...µ s ( x ′ ) = Q ( s ) Z γ dτ δ ( x, x ′ ) (cid:16) u µ . . . u µ s − s g ( µ µ u µ . . . u µ s ) (cid:17) − double traces . (45) B. Solution of the field equations in R Let us now solve the field equations (45). We begin with the R case. The worldline γ isnow just a straight line, which we can think of as running along the (Euclidean) time direc-tion. Let us extend the 4-velocity u µ into a constant vector field in spacetime, which we’ll15enote as t µ . Let R denote our distance from the worldline, and let r µ be the correspondingradius-vector (we denote these by different-case letters, for consistency with the curved casebelow). We then have the basic identities: t µ t µ = 1 ; r µ r µ = R ; t µ r µ = 0 ; ∇ µ t ν = 0 ; ∇ µ R = r µ R ; ∇ µ r ν = q µν , (46)where q µν = g µν − t µ t ν is the flat metric of the 3d space orthogonal to t µ .
1. The solution
For spin 0, the field equation (45) and its solution read: (cid:3) h = Q (0) δ ( r ) ; h ( x ) = − Q (0) πR . (47)For nonzero spins, we will work in a gauge that is traceless (but not transverse). The fieldequation (45) then reads: (cid:3) h µ ...µ s − s ∇ ( µ ∇ ν h µ ...µ s ) ν = Q ( s ) δ ( r ) (cid:16) t µ . . . t µ s − s g ( µ µ t µ . . . t µ s ) − double traces (cid:17) . (48)To solve it, we define a null combination of t µ and r µ : k µ ≡ (cid:18) t µ + ir µ R (cid:19) . (49)In Lorentzian signature, this would be a lightlike vector, which defines an affine tangent tothe lightrays emanating from the worldline. In terms of k µ and r µ , eqs. (46) become: k µ k µ = 0 ; r µ r µ = R ; k µ r µ = iR ∇ µ k ν = i Ω µν R ; ∇ µ R = r µ R ; ∇ µ r ν = q µν , (50)where Ω µν = q µν − R r µ r ν is the metric of the 2-sphere at radius r . In terms of k µ and r µ ,the metrics q µν and Ω µν take the form:Ω µν = g µν − k µ k ν + 4 iR k ( µ r ν ) ; q µν = Ω µν + r µ r ν R . (51)We now claim that the following Kerr-Schild-like field, familiar from [15], solves the fieldequation (48) for all nonzero spins s ≥ h µ ...µ s ( x ) = − Q ( s ) πR k µ . . . k µ s . (52)16ote that this differs by a factor of 2 from the s = 0 case (47). The solution (52) is tracelessas promised, since k µ is null. Plugging it into the field equation (48), one can easily verifythat the LHS vanishes at R = 0, using the following corrolaries of eqs. (50)-(51): ∇ µ k µ = iR ; k ν ∇ ν k µ = r ν ∇ ν k µ = 0 ; (cid:3) R = 0 ; (cid:3) k µ = − ir µ R ; ∇ ρ k µ ∇ ρ k ν = − R (cid:18) g µν − k µ k ν + ik ( µ r ν ) R (cid:19) . (53)What remains is to resolve the delta-function-like source at R = 0. For that purpose, wewrite the Fronsdal tensor on the LHS of (48) as a total divergence: F µ ...µ s = ∇ ν K νµ ...µ s ; K νµ ...µ s = ∇ ν h µ ...µ s − s ∇ ( µ h νµ ...µ s ) . (54)We now need to show that the flux of K ν µ ...µ s through a 2-sphere at radius R reproducesthe coefficient of the delta function on the RHS of (48):4 πR h r ν K ν µ ...µ s i S = Q ( s ) (cid:16) t µ . . . t µ s − s g ( µ µ t µ . . . t µ s ) − double traces (cid:17) . (55)As a first step, we note that r ν K νµ ...µ s is double-traceless already before the S averaging,thanks to the tracelessness of h µ ...µ s . When evaluated explicitly, it reads: r ν K νµ ...µ s = Q ( s ) πR (cid:18) k µ . . . k µ s − is R r ( µ k µ . . . k µ s ) − s ( s − ( µ µ k µ . . . k µ s ) (cid:19) = Q ( s ) πR (cid:18) k µ . . . k µ s − is R r ( µ k µ . . . k µ s ) + s ( s − R r ( µ r µ k µ . . . k µ s ) − s ( s − q ( µ µ k µ . . . k µ s ) (cid:19) . (56)The double-tracelessness is manifest in the first line, since k µ is null and orthogonal to Ω µν .To perform the S average, we decompose (56) along t µ , q µν and r µ : r ν K νµ ...µ s = s ! Q ( s ) s +1 π s X n =0 i n (1 − n ) r ( µ . . . r µ n t µ n +1 . . . t µ s ) n !( s − n )! R n +1 − s − X n =0 i n r ( µ . . . r µ n q µ n +1 µ n +2 t µ n +3 . . . t µ s ) n !( s − n − R n +1 ! . (57)The S averaging now affects only the r µ . . . r µ n factors. For odd n , these average to zero,while for even n , we have the identity: h r µ . . . r µ n i S = R n n + 1 q ( µ µ . . . q µ n − µ n ) . (58)17he latter can be proved by contracting both sides with q µ µ . . . q µ n − µ n , and using theidentity: q µ µ . . . q µ n − µ n q ( µ µ . . . q µ n − µ n ) = n + 1 , (59)which is easy to prove recursively in n . Plugging (58) into (57), we find that the two termsin (57) combine nicely, giving: h r ν K νµ ...µ s i S = Q ( s ) s +1 πR ⌊ s/ ⌋ X n =0 ( − n (cid:18) s n (cid:19) q ( µ µ . . . q µ n − µ n t µ n +1 . . . t µ s ) . (60)To compare with (55), we must re-express the sum in (60) in terms of g µν and t µ , bysubstituting q µν = g µν − t µ t ν . Since the double-tracelessness is assured, it’s enough tocompare the coefficients of t µ . . . t µ s and of g ( µ µ t µ . . . t µ s ) . This is easy to do, confirmingthe flux relation (55), and with it the field equation (48).
2. Field strength and symmetric gauge
We can now calculate the Weyl-like field strength (11) of the solution (52): ϕ µ ν ...µ s ν s ( x ) = − (2 s )! s ! · Q ( s ) S ⊥ µ ν . . . S ⊥ µ s ν s πR s +1 − traces , (61)where S ⊥ µν is a bivector in the tr plane: S ⊥ µν ≡ t [ µ r ν ] = 2 k [ µ r ν ] , (62)and the “ ⊥ ” superscript is in anticipation of the EAdS case below. The derivation of thefield strength (61) from the potential (52) is easy, once it is organized in terms of S ⊥ µν . Therelevant identities read: k [ µ ∇ ν ] k ρ = S ⊥ µν k ρ R + i R k [ µ g ν ] ρ ; ∇ µ S ⊥ νρ = t [ ν g ρ ] µ , (63)where every term proportional to g µν can be discarded as a trace piece. Note that, unlikethe potential (52), the field strength (64) correctly covers also the spin-0 case (47). Thesubtraction of traces in (61) can again be expressed as a projection onto the purely right-handed and purely left-handed parts: ϕ µ ν ...µ s ν s ( x ) = − (2 s )! s ! · Q ( s ) πR s +1 (cid:0) S Lµ ν . . . S Lµ s ν s + S Rµ ν . . . S Rµ s ν s (cid:1) , (64)18here: S R/Lµν ≡ (cid:18) S ⊥ µν ± ǫ µν ρσ S ⊥ ρσ (cid:19) . (65)Finally, we note the freedom of gauge-transforming the solution (52). One reason to prefera different gauge is that (52) discriminates between the two null vectors k µ = ( t µ + iR r µ )and ¯ k µ = ( t µ − iR r µ ), thus breaking time-reversal symmetry t → − t (in the context of blackholes, this can be natural, if one wishes to ignore the time-reversed white hole). It’s easyto see that if we complex-conjugate the solution (52), i.e. replace k µ → ¯ k µ everywhere, theFronsdal tensor (48) and Weyl-like field strength (64), which are real, remain unchanged.Therefore, replacing k µ → ¯ k µ is a gauge transformation. Taking the average of (52) and itscomplex conjugate, we obtain the solution in a gauge that is real and symmetric under timereversal: h µ ...µ s ( x ) = − Q ( s ) πR (cid:0) k µ . . . k µ s + ¯ k µ . . . ¯ k µ s (cid:1) = − Q ( s ) πR Re( k µ . . . k µ s ) . (66)For s = 1, this is the usual Coulomb potential − Q (1) t µ / (4 πR ). C. Solution of the field equations in
EAdS Now, consider the same field equation with a particle source (45) in
EAdS , where weagain work in the embedding-space formalism. The source particle’s geodesic worldline nowstretches from one point ℓ ′ µ to another ℓ µ on the EAdS boundary (see figure 1a). In termsof these, the worldline and its 4-velocity are given by: x µ ( τ ) = 1 √− ℓ · ℓ ′ (cid:0) e τ ℓ µ + e − τ ℓ ′ µ (cid:1) ; u µ ( τ ) = 1 √− ℓ · ℓ ′ (cid:0) e τ ℓ µ − e − τ ℓ ′ µ (cid:1) . (67)To each spacetime point x µ ∈ EAdS away from the worldline, we can again associate aradial parameter R , a radial vector r µ and a “time” vector t µ : R = r − x · ℓ )( x · ℓ ′ ) ℓ · ℓ ′ − r µ = x µ + 12 (cid:18) ℓ µ x · ℓ + ℓ ′ µ x · ℓ ′ (cid:19) ; t µ = 12 (cid:18) ℓ ′ µ x · ℓ ′ − ℓ µ x · ℓ (cid:19) . (69) r µ and t µ lie in the EAdS tangent space at x , i.e. x µ r µ = x µ t µ = 0. The variables ( R, r µ , t µ )play similar roles to their flat counterparts from section III B, and coincide with them close19 a) (b) FIG. 1: (a) An HS-charged particle traveling along a geodesic between two boundary points ℓ, ℓ ′ in EAdS is generating HS gauge fields at a bulk point x . (b) A Feynman diagram in the boundaryvector model, connecting a bilocal operator ¯ φ I ( ℓ ′ ) φ I ( ℓ ) to a local current at x , which can be thoughtof as a boundary limit of the bulk point x ; the solid lines are propagators, while the dashed linesimply contracts the color index I . The boundary diagram in (b) can be viewed as an HS multipletof OPE blocks, between the two fundamental fields φ I ( ℓ ) , ¯ φ I ( ℓ ′ ) and the currents j ( s ) ( x ) of allspins, which constitute the full OPE of φ I ( ℓ ) and ¯ φ I ( ℓ ′ ). The bulk picture in (a), with the particleassigned the DV pattern of charges, is a geodesic Witten diagram for these OPE blocks. to the worldline, where the curvature of EAdS can be neglected. In particular, R is ameasure of distance from the worldline, r µ is tangent to the radial geodesics that emanatefrom the worldline perpendicularly, and t µ is an extension of the 4-velocity u µ from theworldline into the rest of spacetime. The identities (46) acquire curvature corrections, as: t µ t µ = 11 + R ; r µ r µ = R R ; t µ r µ = 0 ; ∇ µ t ν = − t ( µ r ν ) ; ∇ µ R = 1 + R R r µ ; ∇ µ r ν = g µν − t µ t ν − r µ r ν . (70)We again define a null combination k µ of t µ and r µ , according to eq. (49). In Lorentzian,this would again be an affine tangent to the lightrays emanating from the worldline. The20dentities (50)-(53) with curvature corrections read: k µ k µ = 0 ; k µ r µ = iR R ) ; ∇ µ k ν = i R g µν − iR k µ k ν − R ) R k ( µ r ν ) ; ∇ µ r ν = g µν − k µ k ν + 4 iR k ( µ r ν ) + 1 − R R r µ r ν ; ∇ µ k µ = iR ; k ν ∇ ν k µ = 0 ; r ν ∇ ν k µ = − R k µ R ; (cid:3) R = − R ; (cid:3) k µ = − k µ − i (1 + R ) R r µ ; ∇ ρ k µ ∇ ρ k ν = − R g µν + 1 + R R k µ k ν − i (1 + R ) R k ( µ r ν ) . (71)We also define a bivector in the tr plane, as in (62) but with a curvature-corrected prefactor: S ⊥ µν ≡ (1 + R ) t [ µ r ν ] = 2(1 + R ) k [ µ r ν ] , (72)which satisfies identities very similar to (63): k [ µ ∇ ν ] k ρ = S ⊥ µν k ρ R + i R k [ µ g ν ] ρ ; ∇ µ S ⊥ νρ = (1 + R ) t [ ν g ρ ] µ , (73)and can be decomposed into left-handed and right-handed parts as: S L/Rµν ≡ (cid:18) S ⊥ µν ± ǫ µν λρσ x λ S ⊥ ρσ (cid:19) . (74)With these building blocks in hand, it’s easy to show that the solution to the field equation(45) in EAdS , as well as its Weyl curvature, take the same form as in the flat case: h ( x ) = − Q (0) πR ( s = 0) ; (75) h µ ...µ s ( x ) = − Q ( s ) πR k µ . . . k µ s ( s ≥
1) ; (76) ϕ µ ν ...µ s ν s ( x ) = − (2 s )! s ! · Q ( s ) πR s +1 (cid:0) S Lµ ν . . . S Lµ s ν s + S Rµ ν . . . S Rµ s ν s (cid:1) ( s ≥ . (77)To verify this, one must only repeat the calculations at R = 0. The analysis of the fieldequation at the R = 0 singularity can be taken directly from the flat case, since the constantcurvature of EAdS becomes irrelevant at very short distances. For the s ≥ h µ ...µ s ( x ) = − Q ( s ) πR Re( k µ . . . k µ s ) . (78)21inally, it will be useful to express S ⊥ µν directly in terms of the worldline’s boundary endpoints ℓ µ , ℓ ′ µ and the bulk “measurement point” x µ . It turns out that S ⊥ µν is just the projection ofthe bivector ℓ [ µ ℓ ′ ν ] / ( ℓ · ℓ ′ ) into the tangent space at x , i.e. in perpendicular to x µ : S µν ≡ ℓ [ µ ℓ ′ ν ] ℓ · ℓ ′ ; S ⊥ µν = S µν + x µ x ρ S ρν + x ν x ρ S µρ . (79) IV. HS INTERACTION BETWEEN TWO BULK PARTICLES
In this section, we study the action for two bulk particles with geodesic worldlines, in-teracting at leading order via HS fields. We will see that a particular pattern of chargesleads to divergence cancellation and an especially simple result for the action. In section VI,we will identify this special pattern of charges as that of the Didenko-Vasiliev solution, andassociate it with a bilocal operator on the boundary.
A. General structure of the on-shell action
In general, when we impose the field equations G µ ...µ s = T µ ...µ s , the two terms in theaction (39) become proportional to each other. The on-shell action then reads simply: S = 12 Z d x √ g ∞ X s =0 h µ ...µ s T µ ...µ s + boundary terms , (80)where we included a sum over spins. Specializing to a point-particle source, charged underthe fields of different spins, this becomes: S = 12 Z γ dτ ∞ X s =0 Q ( s ) h µ ...µ s u µ . . . u µ s + boundary terms . (81)We will consider here two particles, so really there should be a sum over two worldlines in(81). However, we’ll restrict our attention to the action due to the fields of one particleacting on the other . As usual, there will be an equal contribution from the second particleacting on the first; taking this into account cancels the factor of in (81). The action ofa particle’s fields on itself, i.e. the particle’s self-interaction, is typically UV-divergent. Aswe will see below, this is actually not the case for an HS-charged particle with the Didenko-Vasiliev pattern of charges. Nevertheless, we’ll treat that as just a special case of one particleacting on another. 22wo further subtelties should be addressed before the action (81) can be evaluated:the action’s boundary terms, and its gauge-dependence. The boundary terms in (81) areassociated with the free-field part of the action, and have been with us since eq. (9). Aswe saw in section II B, in EAdS they amount to an integral of the form (18) in transversetraceless gauge. Since neither of the gauges (76),(78) is transverse, this formula cannot beapplied directly. However, we know that the boundary data A i ...i s and J i ...i s in transversetraceless gauge are associated with the magnetic and electric parts (16)-(17) of the gauge-invariant field strength. In other words, the boundary action (18) couples the boundarymagnetic field to the boundary electric field (where the magnetic potential A i ...i s is neededto make this coupling local). Now, it’s straightforward to check that, away from the particle’sworldline, the field strength (77) satisfies B i ...i s = 0, i.e. its asymptotics is purely electric(this was demonstrated in [20], using the language of sections V-VI below). This leavesthe possibility of contributions to B i ...i s from the wordline endpoints ℓ, ℓ ′ ; however, one canrule these out by SO (3) rotational symmetry, after using the boundary conformal group toplace ℓ, ℓ ′ at opposite poles of the boundary S . Thus, the magnetic boundary field stengthvanishes everywhere, and with it the boundary action (18). Having thus concluded that theaction’s boundary terms in EAdS vanish, we will ignore them in the R case as well, sincethe main purpose of the latter is to serve as a toy version for the EAdS calculation.The second issue is gauge dependence. As is already the case in electromagnetism, thecoupling of the bulk particle to HS gauge fields is gauge-invariant only up to boundaryterms, i.e. up to contributions at the worldline endpoints. A natural resolution to thisissue is to restrict to gauges that respect the spacetime symmetries of the source particle’sworldline: the SO (3) of rotations, the R of translations along the worldline, and the Z ofinterchanging the worldline’s endpoints (while also reversing all odd-spin charges). Undera gauge transformation within this class of gauges, the contributions to the action fromthe two endpoints of a worldline will always cancel. Therefore, imposing these spacetimesymmetries fixes the action’s gauge-dependence.To summarize, we’ll be evaluating an action of the form: S = 12 Z γ dτ ∞ X s =0 Q ( s )2 u µ . . . u µ s h µ ...µ s ( x ; γ ) , (82)with no boundary terms, where the integral is over the worldline of particle no. 2, and thefield h µ ...µ s ( x ; γ ) is the one generated by particle no. 1 at the location of particle no. 2.23oreover, we will use h µ ...µ s in a gauge that respects the symmetries of particle no. 1’sworldline. Such a gauge is provided by eq. (78), with the spin-0 case given separately byeq. (75). Plugging these in, the index contractions in (82) reduce to powers of the scalarproduct k µ u µ , and the action takes the form: S = − π Z γ dτR Q (0)1 Q (0)2 + Re ∞ X s =1 Q ( s )1 Q ( s )2 ( k µ u µ ) s ! = − π Z γ dτR Q (0)1 Q (0)2 + Re ∞ X s =1 Q ( s )1 Q ( s )2 s (cid:18) t µ u µ + ir µ u µ R (cid:19) s ! . (83)Here and below, the distance R , the radial vector r µ , the “time” vector t µ and their nullcombination k µ are defined with respect to worldline no. 1, and evaluated at the location ofwordline no. 2; the 4-velocity u µ is that of worldline no. 2. B. The R case We begin in flat Euclidean spacetime. The two particles’ worldlines are straight lines,at distance b and angle θ (in the Lorentzian case, these would describe the particles’ im-pact parameter and relative velocity). We can align the coordinate axes such that the twoworldlines are situated at: x µ ( τ ) = ( τ, , ,
0) ; x µ ( τ ) = ( τ cos θ, τ sin θ, b, . (84)The geometric ingredients of the action formula (83) then read: t µ = (1 , , ,
0) ; r µ = (0 , τ sin θ, b,
0) ; R = p τ sin θ + b ; u µ = (cos θ, sin θ, ,
0) ; t µ u µ = cos θ ; r µ u µ = τ sin θ , (85)so that the action takes the form: S = − π Z ∞−∞ dτ √ τ sin θ + b Q (0)1 Q (0)2 + Re ∞ X s =1 Q ( s )1 Q ( s )2 s (cid:18) cos θ + iτ sin θ √ τ sin θ + b (cid:19) s ! . (86)This integral is scale-invariant. Upon switching to a dimensionless integration variableˆ τ ≡ τ /b , the b -dependence disappears: S = − π Z ∞−∞ d ˆ τ √ ˆ τ sin θ + 1 Q (0)1 Q (0)2 + Re ∞ X s =1 Q ( s )1 Q ( s )2 s (cid:18) cos θ + i ˆ τ sin θ √ ˆ τ sin θ + 1 (cid:19) s ! . (87)24n the other hand, the integral is generally divergent as ˆ τ → ±∞ . For parallel or anti-parallel worldlines, i.e. θ = 0 , π , we get a linear divergence: S = − πb Q (0)1 Q (0)2 + Re ∞ X s =1 (cid:18) ± (cid:19) s Q ( s )1 Q ( s )2 ! Z ∞−∞ dτ , (88)where the ± signs are for θ = 0 , π respectively, and for the moment we restored the dimen-sionful variables τ, b . The general form of eq. (88) is easy to understand: two particles atrest have some potential energy of interaction that scales as inverse distance ∼ /b , anddefines the action per worldline length . Now, there exist particular combinations of charges Q ( s )1 , for which the coefficient in parentheses in (88) vanishes; for these special combinations,the potential energy (and the resulting force) between two particles at rest is zero. Themost famous example is a pair of extremal charged black holes in Einstein-Maxwell theory,or BPS particles in supergravity. For two such objects at rest, the electric repulsion preciselycancels the gravitational attraction. In our notation, this corresponds to the case θ = 0,with particles charged only under the s = 1 gauge field (electric charge) and the s = 2field (gravitational mass), with the charges related as Q (1) = ± i √ Q (2) . Here, the imaginaryelectric charge is a standard consequence of working in Euclidean signature. At θ = π , thesesame charges yield contributions that add up rather than cancel; in Lorentzian signature,this corresponds to a particle and anti particle at rest, with both electric and gravitationalforces attractive.The integral (87) diverges also for general angles 0 < θ < π , but logarithmically ratherthan linearly. Specifically, at both ends τ → ±∞ of the worldline, the integral takes theform: S log-divergent = − π sin θ Q (0)1 Q (0)2 + ∞ X s =1 Q ( s )1 Q ( s )2 s cos( sθ ) ! Z dτ | τ | . (89)Comparing with (88), we can see that the introduction of an angle (i.e. a relative velocity)brings out the different tensor structures of the different-spin interactions, in the form ofthe angle-dependent cos( sθ ) factors. In particular, the s = 1 and s = 2 contributions to thelogarithmic divergence have different θ dependence, and therefore can no longer cancel eachother; in particular, for the BPS charge assignment Q (1) = i √ Q (2) , we get a cancellationonly at θ = 2 π/ s = 1 ,
2, we can obtain acancellation of the divergences at almost all angles . Suppose, as in the BPS case, that both25articles have the same proportionality pattern between the charges of different spins, i.e.: Q ( s )1 Q (0)1 = Q ( s )2 Q (0)2 ≡ q s , (90)The logarithmic divergence (89) of the action then reads: S log-divergent = − Q (0)1 Q (0)2 π sin θ
12 + ∞ X s =1 q s s cos( sθ ) ! Z dτ | τ | . (91)We see that the squared charges of the different spins act as Fourier coefficients for the θ -dependence of the logarithmic divergence. This means that we cannot quite cancel thedivergence for all θ , but we can increase the domain of cancellation from θ = 0 all the wayto 0 ≤ θ < π , by making the expression in parentheses in (91) proportional to δ ( θ − π ):12 + ∞ X s =1 q s s cos( sθ ) ∼ δ ( θ − π ) . (92)This is accomplished by choosing q s = ( − s , i.e.: Q ( s ) = ± ( i √ s Q (0) , (93)which is consistent with the BPS assignment for s = 1 ,
2, but extends it to all spins. As wewill see in section VI, the pattern of charges (93) coincides with the one for the Didenko-Vasiliev black hole. We therefore refer to it as the DV pattern. Plugging it back into the fullaction formula (87), we find that the sum over spins becomes a geometric series. Summingthe series, we arrive at an integral that can be performed analytically: S = − Q (0)1 Q (0)2 (1 − cos θ )8 π (1 + cos θ ) Z ∞−∞ d ˆ τ (cid:0) τ (1 − cos θ ) + 1 (cid:1) √ ˆ τ sin θ + 1= − Q (0)1 Q (0)2 π (1 + cos θ ) arctan (1 − cos θ )ˆ τ √ ˆ τ sin θ + 1 (cid:12)(cid:12)(cid:12)(cid:12) ∞ ˆ τ = −∞ = − Q (0)1 Q (0)2 θ π (1 + cos θ ) . (94)As anticipated above, we see that the action vanishes for θ = 0, diverges for θ = π , and isfinite for all intermediate values 0 < θ < π . To recapitulate, this finiteness is due to can-cellations of the divergences (88)-(89), in a higher-spin-enhanced version of the cancellationfor BPS particles in supergravity, which takes place only at θ = 0 and θ = 2 π/ b dropped out of theaction (94), due to scale invariance. This scale invariance enables us to take two equivalent26 a) (b) FIG. 2: (a) Two HS-charged particles, traveling along geodesics in
EAdS , are interacting via theirHS gauge fields. (b) A Feynman diagram in the boundary vector model, computing the correlatorof two bilocal operators ¯ φ I ( ℓ ′ ) φ I ( ℓ ) and ¯ φ I ( ℓ ′ ) φ I ( ℓ ); the solid lines are propagators, while thedashed lines are just contractions of the color indices I . The bulk picture in (a), with each of thetwo particles assigned the DV pattern of charges, describes an HS multiplet of Witten diagramsthat compute the same correlator. The boundary diagram in (b) is almost one of the Feynmandiagrams for the 4-point function of scalar operators j (0) ( ℓ ) = ¯ φ I ( ℓ ) φ I ( ℓ ), but with two propagatorsmissing. viewpoints on the cancelled divergences. Our calculation above made them appear as IRdivergences: at fixed impact parameter b , the action diverges as we integrate over distantportions of the worldline. However, these same divergences can be viewed as UV ones: if wecut off the integration at some fixed distance along the worldline, then the action becomesfinite at fixed b , but diverges for b →
0, i.e. when the two particles collide.
C. The
EAdS case We now turn to the case of two particles interacting via HS fields in
EAdS (see figure2a). Our first task is again to parameterize the particles’ geodesic worldlines. The relativeposition of the worldlines is again characterized by two numbers: an impact parameter,which we will now denote by χ , and a relative angle θ . χ is defined as the length of theshortest interval in EAdS connecting the two worldlines; it is a hyperbolic angle in the R , embedding space. θ is defined as the angle between the worldlines’ 4-velocities, evaluated atthe ends of this shortest interval (the angle can be defined equivalently either in embeddingspace, or intrinsically in EAdS using parallel transport along the interval).27n a suitably chosen Lorentz frame in the R , embedding space, we can fix the positionsand 4-velocities of the two worldlines at their closest points as: x µ (0) = (1 , , , ,
0) ; u µ (0) = (0 , , , ,
0) ; x µ (0) = (cosh χ, , sinh χ, ,
0) ; u µ (0) = (0 , cos θ, , sin θ, . (95)From these, we can construct the worldlines themselves as: x µ ( τ ) = x µ (0) cosh τ + u µ (0) sinh τ = (cosh τ, sinh τ, , ,
0) ; (96) x µ ( τ ) = x µ (0) cosh τ + u µ (0) sinh τ = (cosh χ cosh τ, cos θ sinh τ, sinh χ cosh τ, sin θ sinh τ, . (97)In particular, the 4-velocity along the 2nd worldline reads: u µ ( τ ) ≡ u µ ( τ ) = (cosh χ sinh τ, cos θ cosh τ, sinh χ sinh τ, sin θ cosh τ, . (98)Taking the limits τ → ±∞ in (96)-(97) and extracting coefficients of e | τ | , we identify theboundary endpoints of the two worldlines as: ℓ µ = (1 , , , ,
0) ; ℓ ′ µ = (1 , − , , ,
0) ; (99) ℓ µ = (cosh χ, cos θ, sinh χ, sin θ,
0) ; ℓ ′ µ = (cosh χ, − cos θ, sinh χ, − sin θ, . (100)Our two parameters χ and θ are just a particular encoding of the two independent cross-ratios of the four boundary points ℓ , ℓ ′ , ℓ , ℓ ′ . Explicitly: s ( ℓ · ℓ ′ )( ℓ · ℓ ′ )( ℓ · ℓ ′ )( ℓ · ℓ ′ ) = 12 (cosh χ + cos θ ) ; s ( ℓ · ℓ )( ℓ ′ · ℓ ′ )( ℓ · ℓ ′ )( ℓ · ℓ ′ ) = 12 (cosh χ − cos θ ) . (101)Plugging the 2nd worldline’s position (97) and the 1st worldline’s endpoints (99) into eqs.(68)-(69), we obtain the ingredients of the action integral (83) as: R = q (cosh χ − cos θ ) sinh τ + sinh χ ; r µ u µ = (cosh χ − cos θ ) cosh τ sinh τ (cosh χ − cos θ ) sinh τ + cosh χ ; t µ u µ = cosh χ cos θ (cosh χ − cos θ ) sinh τ + cosh χ . (102)28e can simplify these expressions somewhat by switching variables from τ (which rangesfrom −∞ to ∞ ) to R (which ranges twice from sinh χ to ∞ ). Plugging everything into theaction integral (83), we get: S = − π Z ∞ sinh χ dR q ( R − sinh χ )( R + sin θ ) Q (0)1 Q (0)2 ++ Re ∞ X s =1 Q ( s )1 Q ( s )2 s ( R + 1) s (cid:18) cosh χ cos θ + iR q ( R − sinh χ )( R + sin θ ) (cid:19) s ! , (103)The integral (103) does not have the scale-invariance of its R counterpart, due to the EAdS curvature radius (here, set to 1). Due to the negative curvature, geodesics in EAdS recede from each other at large distances much faster than in R . As a result, unlike its flatcounterpart, the integral (103) is IR-finite for any assignment of charges Q ( s )1 , : at R → ∞ ,the spin- s piece of the integral goes as R dR/R s +2 . On the other hand, there are still UVdivergences in the limit χ →
0, i.e. as the two particles collide. Since the spacetime curvaturebecomes negligible at short distances, these UV divergences are the same as the ones westudied in the R case. They can thus be cancelled, for all values of θ except θ = π , byassigning the DV pattern of charges (93) to both particles. With this assignment, the sumover spins in (103) again becomes a geometric sum. Performing it, we bring the actionintegral into the form: S = Q (0)1 Q (0)2 π Z ∞ sinh χ dR q ( R − sinh χ )( R + sin θ ) −− ( R + 1) Re (cid:18) R + 1 + cosh χ cos θ + iR q ( R − sinh χ )( R + sin θ ) (cid:19) − ! . (104)We’ve been unable to significantly simplify this integral, or to evaluate it analytically. Onemight have hoped to at least use the flat result (94) in the limit χ → EAdS result isn’t captured by the R one, precisely due to the cancellation of UV divergences. Onthe other hand, as we’ll discuss below, holography predicts a very simple answer for theintegral (104): S = − Q (0)1 Q (0)2 χ + cos θ ) = − Q (0)1 Q (0)2 s ( ℓ · ℓ ′ )( ℓ · ℓ ′ )( ℓ · ℓ ′ )( ℓ · ℓ ′ ) . (105)29e’ve verified the agreement between (104) and (105) by numerical integration in Mathe-matica, for various values of the parameters χ, θ . The formula (105) for the leading-orderinteraction of two HS-charged particles in EAdS with the DV pattern of charges is the maintechnical result of our paper. In the next sections, we will place our analysis of HS-chargedparticles in a broader context, by connecting it to the Didenko-Vasiliev black hole solution,as well as to higher-spin holography. V. TWISTORS, HS ALGEBRA AND BOUNDARY CORRELATORS
From here on, we focus on the case of
EAdS spacetime. In this section, we introducethe tools necessary to connect our results above with the Didenko-Vasiliev solution and withAdS/CFT. In essence, we need to switch from the Fronsdal’s “metric-like” formulation of HSfields to Vasiliev’s language of twistors, HS algebra and master fields. More specifically, wewill introduce these in a slightly non-standard approach, developed by us in [20, 21, 25, 33].The idea is to work in the embedding-space picture, and to introduce twistors in a waythat is closer to Penrose’s original sense of the word [34, 35] – as spinors of the spacetimesymmetry group SO (1 , EAdS , and the HS-algebraic generating function for its correlators. In sectionV C, we use the boundary 2-point functions to fix the relative normalizations between theFronsdal and twistor languages. The content of sections V A-V B is a telegraphic summaryof constructions detailed at length in [20, 21]; the calculation in section V C is new. A. Twistor space, HS algebra and the Penrose transform
For the purposes of this paper, twistor space is the space of (4-component, Dirac) spinorsof the
EAdS isometry group SO (1 , R , embedding space. We use Latin indices ( a, b, . . . ) to denote twistors. Twistor space hasa symplectic metric I ab with inverse I ab I ac = δ bc , which we use to raise and lower indicesas U a = I ab U b , U a = U b I ba . It is often convenient to use index-free notation, in whichbottom-to-top index contraction is implied, e.g. U V ≡ U a V a . The translation between30wistors and tensors is performed by the Dirac gamma matrices ( γ µ ) ab , which satisfy theClifford algebra γ ( µ γ ν ) = − η µν (these are just the familiar gamma matrices from R , , withthe addition of γ ). It is also useful to define the antisymmetric combinations γ µν ≡ γ [ µ γ ν ] ,which generate the SO (1 ,
4) spacetime symmetry within Clifford algebra. The matrices γ abµ are antisymmetric and traceless in their twistor indices, while the γ abµν are symmetric. Wedefine the following dictionaries between objects with tensor and twistor indices: ξ ab = γ abµ ξ µ ; ξ µ = − γ µab ξ ab ; ζ ab = 12 γ abµν ζ µν ; ζ µν = 14 γ µνab ζ ab . (106)We now define HS algebra in complete analogy to Clifford algebra. Instead of a vectorof quantities γ µ whose anticommutator is given by the spacetime metric η µν , we define a twistor variable Y a whose commutator is given by the twistor metric I ab . We denote thisnon-commutative product with a star ⋆ , and realize it as: Y a ⋆ Y b = Y a Y b + iI ab . (107)The can be extended into an associative product on twistor functions f ( Y ). For generalfunctions, it is given by an integral formula: f ( Y ) ⋆ g ( Y ) = Z d U d V f ( Y + U ) g ( Y + V ) e − iUV , (108)where the twistor integration measure is defined as: d U ≡ ǫ abcd π ) dU a dU b dU c dU d ; ǫ abcd ≡ I [ ab I cd ] . (109)The algebra defined by the product (107)-(108) (restricted to even functions, i.e. to integerspins) is known as HS algebra, and defines the infinite-dimensional symmetry group of HStheory. It contains within itself the spacetime symmetry SO (1 , Y a Y b . HS algebra admits a trace operation,defined simply by: tr ⋆ f ( Y ) = f (0) . (110)So far, we made no reference to any spacetime points. Choices of spacetime points, eitherin the bulk of EAdS or on its boundary, induce decompositions of twistor space, and thusof HS algebra. A choice of bulk point x µ decomposes twistor space into right-handed andleft-handed Weyl spinor spaces at x , via the projectors: P ab ( ± x ) = 12 (cid:0) I ab ± x µ γ abµ (cid:1) , (111)31r, in index-free notation, simply P ( ± x ) = (1 ± x ). We use the same notation P ( ± x )to denote the two spinor spaces themselves. The two spinor spaces at x are orthogonal toeach other under the twistor metric, which simply decomposes as I ab = P ab ( x ) + P ab ( − x ).We denote the right-handed and left-handed Weyl-spinor pieces of a twistor U a at x as u ( ± x ) ≡ P ( ± x ) U . We define a measure on each spinor space, and a corresponding deltafunction, as: d u ( ± x ) = P ab ( ± x )2(2 π ) du a ( ± x ) du b ( ± x ) ; δ ± x ( Y ) = Z P ( ± x ) d u e iuY . (112)A key concept in twistor theory is the Penrose transform , which maps between twistorfunctions and solutions to free massless field equations in 4d. It was noticed in [20] thatthe Penrose transform has a very elegant expression in HS theory. Specifically, we can mapbetween an even twistor function f ( Y ) and a master field C ( x ; Y ) in EAdS that containsan HS multiplet of free massless fields, via a simple star product: C ( x ; Y ) = if ( Y ) ⋆ δ x ( Y ) . (113)When written out explicitly, the star product in (113) is a Fourier transform of the right-handed spinor y ( x ) (the Penrose transform is famously chiral; of course, a left-handed trans-form can also be defined). The master field C ( x ; Y ) = C ( x ; y ( x ) + y ( − x ) ) acts as a generatingfunction for the Weyl-like field strengths (11) of HS fields of all spins, together with theirderivatives. The field strengths themselves (as opposed to their derivatives) are containedin the purely chiral parts C ( x ; y ( x ) ) and C ( x ; y ( − x ) ) of the master field, as: C µ ν ...µ s ν s ( x ) = 14 s γ a a µ ν . . . γ a s − a s µ s ν s ∂ s C ( x ; y ( x ) ) ∂y a ( x ) . . . ∂y a s ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ( x ) =0 + ∂ s C ( x ; y ( − x ) ) ∂y a ( − x ) . . . ∂y a s ( − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ( − x ) =0 , (114)where the two terms on the RHS are the right-handed and left-handed parts of the fieldstrength, respectively. The spin-0 field is simply given by: C ( x ) = C ( x ; 0) . (115)The transform (113) automatically ensures that the fields (114)-(115) satisfy the appropriatefield equations (12) in EAdS . Note that our notation for the field strengths (114)-(115)is different from the one we used so far, i.e. ϕ µ ν ...µ s ν s ( x ). This is because we reserve32he latter notation to field strengths derived, via (11), from potentials with a canonicallynormalized kinetic term , as in (9). So far, we haven’t given the twistor function f ( Y ) andthe Penrose-transformed fields (113)-(115) a meaningful normalization. In section V B, wewill equip them with one, by tying them to holographic correlators. In section V C, we willwork out the proportionality coefficients between these fields and the canonically normalizedones from (11). B. Boundary correlators from twistor functions
Here, we begin to turn our attention to the holographic CFT dual of HS gravity, whichlives on the boundary of
EAdS . In the simplest case that we’re considering, this CFT isa free vector model of N complex massless scalar fields φ I , subject to U ( N ) symmetry. Itssingle-trace primary operators are a tower of conserved currents, one for each spin s : j ( s ) k ...k s = 1 i s ¯ φ I s X m =0 ( − m (cid:18) s m (cid:19) ← ∂ ( k . . . ← ∂ k m → ∂ k m +1 . . . → ∂ k s ) − traces ! φ I , (116)whose bulk duals are the HS gauge fields. One can uplift the 3d boundary indices into 5dindices in the R , embedding space. Also, it is convenient to package the tensor componentsof (116) at a boundary point ℓ by contracting with a null polarization vector λ µ , like theone we introduced in section II C (satisfying λ · ℓ = λ · λ = 0): j ( s ) ( ℓ, λ ) = λ µ . . . λ µ s j µ ...µ s ( ℓ )= λ µ . . . λ µ s i s ¯ φ I ( ℓ ) s X m =0 ( − m (cid:18) s m (cid:19) ← ∂ ( µ . . . ← ∂ µ m → ∂ µ m +1 . . . → ∂ µ s ) φ I ( ℓ ) . (117)As discussed in [20, 21], there exists a “holographic dual of the Penrose transform”: a dictio-nary that encodes single-trace operator insertions in the CFT as twistor functions. In turn,these twistor functions correspond via the (ordinary, bulk) Penrose transform to linearizedbulk fields with the appropriate boundary data. In terms of these twistor functions f ( Y ),the generating function for the CFT correlators is given by the HS-algebraic expression: Z [ f ( Y )] = exp N ∞ X n =1 ( − n +1 n tr ⋆ (cid:16) f ( Y ) ⋆ . . . ⋆ f ( Y ) | {z } n factors (cid:17) . (118)This partition function defines the on-shell bulk action of HS gravity (at least in the classicallimit, i.e. at large N ). In particular, it lends meaning to the normalization of the bulk fields(113)-(115) produced from f ( Y ) via the Penrose transform.33o make this more explicit, let us write down the twistor function that corresponds tothe boundary current (117). First, we must briefly discuss the structure imposed on twistorspace by a choice of boundary point ℓ . At a bulk point, we saw that twistor space decomposesinto the two chiral subspaces (111). At a boundary point ℓ , only a single 2d subspace issingled out – the subspace P ( ℓ ) spanned by ℓ ab = ℓ µ γ abµ . This ends up serving as the space of2-component cospinors on the 3d boundary. Though P ( ℓ ) is totally null under the twistormetric I ab , one can equip it with a symplectic metric, or equivalently a measure d u ( ℓ ) , byusing ℓ ab itself: du a ( ℓ ) du b ( ℓ ) π ≡ ℓ ab d u ( ℓ ) . (119)This metric scales under rescalings of ℓ µ , as is appropriate for a metric on the conformalboundary. We can use it to define a delta function with support on P ( ℓ ): δ ℓ ( Y ) = Z d u ( ℓ ) e iu ( ℓ ) Y . (120)The twistor function corresponding to the boundary current (117) is constructed from thisdelta function as [21]: κ ( s ) ( ℓ, λ ; Y ) = iM a . . . M a s π (cid:18) Y a . . . Y a s + ( − s ∂ s ∂Y a . . . ∂Y a s (cid:19) δ ℓ ( Y ) , (121)where M a is a polarization spinor, defined as an appropriate square root of the bivector M µν ≡ ℓ [ µ λ ν ] : γ abµν ℓ µ λ ν = 12 γ abµν M µν = ( ℓM ) a ( ℓM b ) . (122)The Penrose transform (113) of the twistor function (121) reads [21]: C ( x ; Y ) = 14 π · ( M ℓP − x Y ) s + ( M ℓP x Y ) s ( ℓ · x ) s +1 exp iY ℓxY ℓ · x ) . (123)Here, the field strength at x is contained in the exponent’s prefector, while the exponent itselfcarries the tower of derivatives. Explicitly, the field strength, extracted via eqs. (114)-(115),reads: C µ ν ...µ s ν s ( x ) = (2 s )!4 π · M Lµ ν . . . M Lµ s ν s + M Rµ ν . . . M Rµ s ν s ( ℓ · x ) s +1 , (124)where M L/Rµ ν are the projections of M µν onto the left-handed and right-handed bivectorspaces at x , as in (31). The field strength (124) clearly coincides, up to numerical factors,34ith the boundary-to-bulk propagator (32)-(33). The s = 0 case is included in (121) and(124), as: κ (0) ( ℓ ; Y ) = i π δ ℓ ( Y ) ; (125) C ( x ) = 12 π ( ℓ · x ) . (126) C. Fixing the normalization of the Fronsdal/twistor dictionary
Let’s now work out the 2-point function for the spin- s currents (117). By SO (1 , O ( ℓ, ℓ ′ ) = ¯ φ I ( ℓ ′ ) φ I ( ℓ ). Theconnected 2-point function for these is given by the Feynman diagram in figure 2b: hO ( ℓ , ℓ ′ ) O ( ℓ , ℓ ′ ) i connected = N G ( ℓ ′ , ℓ ) G ( ℓ ′ , ℓ ) = N π p ( ℓ ′ · ℓ )( ℓ · ℓ ′ ) . (127)Here, G = (cid:3) − is the propagator of the vector model’s fundamental field φ I : G ( ℓ, ℓ ′ ) = − π √− ℓ · ℓ ′ , (128)which is just the embedding-space expression for the massless propagator G = − / (4 πr ) in3d flat space.To obtain the 2-point function of the spin-0 local “current” j (0) ( ℓ ) = ¯ φ I ( ℓ ) φ I ( ℓ ), we simplyset ℓ = ℓ ′ and ℓ = ℓ ′ in (127): (cid:10) j (0) ( ℓ ) j (0) ( ℓ ) (cid:11) connected = − N π ( ℓ · ℓ ) . (129)For the 2-point function of currents with nonzero spin, we must act on (127) with derivativesaccording to the pattern in (117), contract with polarization vectors λ µ , λ µ , and then set ℓ = ℓ ′ and ℓ = ℓ ′ in the end. Performing this procedure on the first of the two bilocals in(127), we get: (cid:10) j ( s ) ( ℓ , λ ) O ( ℓ , ℓ ′ ) (cid:11) connected = (2 s )!(4 i ) s s ! · N π s X m =0 ( − m (cid:18) sm (cid:19) ( λ · ℓ ) m ( λ · ℓ ′ ) s − m ( − ℓ · ℓ ) m + ( − ℓ · ℓ ′ ) s − m + . (130)35oing the same to the second bilocal, we will get terms of the general form:( λ · λ ) n ( λ · ℓ ) s − n ( λ · ℓ ) s − n ( ℓ · ℓ ) s − n +1 , (131)with coefficients that involve some unpleasant combinatoric sums. On the other hand, weknow that these terms must eventually organize into the structure from (38):( M µν M µν ) s ( ℓ · ℓ ) s +1 = 2 s (( λ · λ )( ℓ · ℓ ) − ( λ · ℓ )( λ · ℓ )) s ( ℓ · ℓ ) s +1 . (132)Thus, it’s enough to follow just the coefficient of e.g. the ( λ · λ ) s / ( ℓ · ℓ ) s +1 term. Thisarises from acting with the ∂/∂ℓ µ and ∂/∂ℓ ′ µ derivatives just on the numerator in (130).The coefficient is now easy to work out, and we get: (cid:10) j ( s ) ( ℓ , λ ) j ( s ) ( ℓ , λ ) (cid:11) connected = ( − s +1 (2 s )! N s +6 π · ( M µν M µν ) s ( ℓ · ℓ ) s +1 . (133)Now, recall that the bulk action is related to the boundary partition function as S = − ln Z .Thus, the quadratic contribution to the bulk action from two boundary insertions, whichcorrespond to the boundary-to-bulk propagators (124), is simply minus the 2-point function(129),(133). To conform with the conventions of the previous sections, we also divide by afactor of 2, so as to count each ordering of the two boundary insertions separately. We thusarrive at the bulk action as: s = 0 : S [ C , C ] = N π ( ℓ · ℓ ) ; s ≥ S [ C , C ] = ( − s (2 s )! N s +7 π · ( M µν M µν ) s ( ℓ · ℓ ) s +1 . (134)On the other hand, for boundary-to-bulk propagators expressed as Fronsdal fields h µ ...µ s ofthe form (28) with curvature ϕ µ ν ...µ s ν s of the form (32)-(33), we’ve seen that the bulk actionis given by eqs. (37)-(38). Putting everything together, we arrive at the proportionalitycoefficients between the field strengths of canonically normalized Fronsdal fields, and thosederived from f ( Y ) via the Penrose transform: C µ ν ...µ s ν s ( x ) = 4 π √ s − N ϕ µ ν ...µ s ν s ( x ) . (135)Eq. (135) holds for both zero and nonzero spins.36 I. THE DIDENKO-VASILIEV SOLUTION AND BOUNDARY BILOCALSA. DV particle as the bulk dual of a boundary bilocal
In section V C, we used the fact that the local single-trace operators (117) of the boundaryCFT can all be treated as singular limits of the simple bilocal operator O ( ℓ, ℓ ′ ) = ¯ φ I ( ℓ ′ ) φ I ( ℓ ).This is the essence of the Flato-Fronsdal theorem [36], which has been highlighted andexploited e.g. in [18, 19]. Now, in [20], we identified the twistor function that correspondsto the bilocal O ( ℓ, ℓ ′ ), in the same sense that the twistor functions (121) correspond to thelocal currents (117). In other words, we found a linear map between bilocal boundary sourcesand twistor functions, such that the correlators hO ( ℓ , ℓ ′ ) . . . O ( ℓ n , ℓ ′ n ) i are generated by theHS-algebraic functional (118). The specific twistor function that corresponds to O ( ℓ, ℓ ′ )reads: K ( ℓ, ℓ ′ ; Y ) = 1 π √− ℓ · ℓ ′ exp iY ℓℓ ′ Y ℓ · ℓ ′ . (136)To understand the origin of this function, we can write it as a star product of two localpieces: K ( ℓ, ℓ ′ ; Y ) = κ (0) ( ℓ ; Y ) ⋆ κ (0) ( ℓ ′ ; Y ) G ( ℓ, ℓ ′ ) = √− ℓ · ℓ ′ π δ ℓ ( Y ) ⋆ δ ℓ ′ ( Y ) , (137)where κ (0) ( ℓ ; Y ) is the twistor function (125) describing a local insertion of the spin-0 operator j (0) ( ℓ ) = ¯ φ I ( ℓ ) φ I ( ℓ ), and G ( ℓ, ℓ ′ ) is the fundamental propagator (128). The logic behind eq.(137) is as follows. The Feynman diagrams of the free vector model’s correlators are simplysingle loops, in which the operator insertions are connected by propagators G ( ℓ, ℓ ′ ) (see e.g.figures 1b and 2b). An O ( ℓ, ℓ ′ ) insertion in such a Feynman diagram behaves exactly like apair of insertions j (0) ( ℓ ) , j (0) ( ℓ ′ ) in sequence, but without the propagator between ℓ and ℓ ′ .Eq. (137) encapsulates this diagrammatic relationship in terms of HS algebra.Now, what is the bulk master field that corresponds to the twistor function (137)? Thiswas also calculated in [20], as: C ( x ; Y ) = ± π p ℓ · ℓ ′ + 2( ℓ · x )( ℓ ′ · x )] exp iY [ ℓℓ ′ + 2( ℓ ′ · x ) ℓx ] Y ℓ · ℓ ′ + 2( ℓ · x )( ℓ ′ · x )] . (138)The overall sign is ambiguous, and will not play an important role. For later convenience,we will set it to −
1. We now wish to make the key observation that the bulk fields contained37n (138) are precisely the fields of an HS-charged particle, moving along the bulk geodesicbetween the boundary points ℓ ′ and ℓ , carrying the Didenko-Vasiliev pattern of charges (93).First, we observe that the denominators in (138) are proportional to the “distance” R fromthe geodesic, as defined in (68): C ( x ; Y ) = − π √− ℓ · ℓ ′ R exp − iY [ ℓℓ ′ + 2( ℓ ′ · x ) ℓx ] Y ℓ · ℓ ′ ) R . (139)Let’s now expand out the compact index-free notation in (139), and highlight the relevantindex symmetries: C ( x ; Y ) = − π √− ℓ · ℓ ′ R exp iY a Y b γ µνab h ℓ [ µ ℓ ′ ν ] + 2( ℓ ′ · x ) ℓ [ µ x ν ] i ℓ · ℓ ′ ) R . (140)Now, recall from (114)-(115) that the field strengths at x (as opposed to their deriva-tives) are contained in the master field’s dependence on purely chiral spinors at x , namely Y = P ± ( x ) Y = y ( ± x ) . When we make this substitution in (140), the bivector in squarebrackets gets projected onto the space of right-handed or left-handed bivectors at x . Sinceboth of these spaces are orthogonal to x µ , the second term in the square brackets can besimply ignored. As for the first term, we recall from eq. (79) that its projection onto thespace of bivectors at x is proportional to the bivector S ⊥ µν in the tr plane, defined in (72).Decomposing this into its right-handed and left-handed parts, we arrive at: C ( x ; y ( ± x ) ) = − π √− ℓ · ℓ ′ R exp iY a Y b γ µνab S R/Lµν R . (141)From here, we extract the field strengths of different spins using (114)-(115): s = 0 : C ( x ) = − π √− ℓ · ℓ ′ · R ; s ≥ C µ ν ...µ s ν s ( x ) = − i s (2 s )! πs ! √− ℓ · ℓ ′ · S Lµ ν . . . S Lµ s ν s + S Rµ ν . . . S Rµ s ν s R s +1 . (142)Finally, we use eq. (135) to convert these into the field strengths of canonically normalizedFronsdal fields: s = 0 : ϕ ( x ) = − √ N π √− ℓ · ℓ ′ · R ; s ≥ ϕ µ ν ...µ s ν s ( x ) = − i s (2 s )! √ s N π s ! √− ℓ · ℓ ′ · S Lµ ν . . . S Lµ s ν s + S Rµ ν . . . S Rµ s ν s R s +1 . (143)38e now observe that these are just the field strengths (75),(77) of an HS-charged particlefrom section III C, with charges: Q ( s ) = i s √ s N π √− ℓ · ℓ ′ . (144)This pattern of charges precisely agrees with the one we identified in eq. (93) as cancellingUV divergences in the two-particle interaction.Note that a curious thing has happened here. Normally, the Penrose transform shouldproduce solutions to free massless field equations, without bulk sources. For the boundary-to-bulk propagators (121),(124) corresponding to the local boundary operator j ( s ) ( ℓ, λ ), thisis indeed the case. However, we now see that the Penrose transform of the twistor function(136) solves not quite the free linearized equations, but the equations with a particle-likesource along the bulk geodesic between ℓ ′ and ℓ . This puts us into somewhat new territoryfor holography. In particular, one may wonder: are the boundary 2-point correlators stilldescribed by a quadratic bulk action, even though the corresponding bulk fields are no longerfree?It turns out that the answer is yes, provided we define the bulk action as in (44), includingboth the free-field term and the interaction term with the bulk “particle”. Indeed, considertwo boundary bilocals, O ( ℓ , ℓ ′ ) and O ( ℓ , ℓ ′ ). Each of these generates bulk HS fields, whichare the fields of an HS-charged particle with charges given by (144). We can then use theresult (105) of section IV to evaluate the quadratic bulk action as: S = − N π p ( ℓ · ℓ ′ )( ℓ · ℓ ′ ) . (145)This is − times the correlator (127) of the two bilocals, in agreement with the holographicdictionary (recall eqs. (129),(133) as compared to eq. (134)). Note that if we were toconsider only the first, “free-field” term in the action (44), the result would have the oppositesign. Thus, the interaction term with the bulk “particle” must be included both in the fieldequations for the linearized HS fields (otherwise (143) is not a solution), and when evaluatingthe bulk action (which otherwise fails to agree with the bilocal correlator (127)). In otherwords, if we wish to work with boundary bilocals, we have no choice but to account for theexistence of DV particle-like sources in the bulk.39 . Relation to the Didenko-Vasiliev “black hole” So far, we’ve shown that the linearized HS fields of a “Didenko-Vasiliev particle”, asdefined in section IV, coincide with the bulk fields that correspond to a bilocal operator inthe boundary CFT. In this section, we observe that they also coincide with the linearizedversion of the Didenko-Vasiliev black hole, thus justifying our nomenculature. Our statementis that the bulk master field (138), derived via the Penrose transform (113) from the twistorfunction (136), is the same as the linearized Didenko-Vasiliev solution as given in [13], upto slight differences in the formalism (and in the spacetime signature).First, let us summarize the differences and similarities between the twistor formalismpresented here and the one found in “mainstream” HS literature. The formalism in thispaper, which was first put forward in [33], starts with a fixed
EAdS geometry, defined viaan R , embedding space. The tangent space at a spacetime point x ∈ EAdS is just thetangent 4d hyperplane in R , to the EAdS hyperboloid; the tangent spaces for differentpoints are represented by different 4d hyperplanes in the same R , . Twistor space is definedas the space of SO (1 ,
4) spinors. At a point x ∈ EAdS , it decomposes into two spaces ofWeyl SO (4) spinors; the Weyl spinor spaces at different points are represented by different2d subspaces of the same twistor space.In contrast, in the standard HS literature, one doesn’t have a fixed EAdS geometry or anembedding space, but an a-priori featureless spacetime manifold. On it, one constructs theframe fields of Cartan’s formulation of General Relativity, and their higher-spin extensions.Thus, the tangent space and Weyl spinor spaces at different spacetime points x exist only asfibers over the spacetime manifold, as is usually the case in GR. The left-handed and right-handed spinors at x are unified into Dirac spinors Y . These are referred to as “twistors”,but “only” due to the structure imposed on them by HS algebra, acting on the fiber at x .There is no notion of a twistor Y that exists independently from the spacetime point x .Again, this is as usual in GR: true, Penrosian, x -independent twistors are easy to defineonly on very special spacetimes.With these basic circumstances in mind, let us consider again the HS-algebraic Penrosetransform (113). The bulk master field C ( x ; Y ) on the LHS of (113) is basically the same asthat in the standard HS literature, up to the aforementioned difference in the nature of Weylspinors at x : in the standard formalism, they are basic structures in the fiber at x , while in40urs, they are x -dependent projections of an x -independent twistor Y . The same commentsapply to the spinor delta function δ x ( Y ) on the RHS of (113). As for the x -independenttwistor function f ( Y ) on the RHS of (113), one may think at first that it has no analog inthe standard HS formalism. And yet, essentially the same formula as (113) was put forwardin eq. (3.23) of [13], as a technique for generating free bulk solutions. Instead of a twistorfunction f ( Y ) that’s literally constant with respect to x , in [13] one uses a function ǫ ( x ; Y )that is covariantly constant with respect to the HS connection, in the adjoint representation of HS symmetry. A star product with a spinor delta function, just as in (113), transformsthis function into a master field that solves the linearized bulk equations, and in turn livesin the so-called twisted adjoint representation of HS symmetry. Upon some reflection, onecan see that eq. (3.23) of [13] and our Penrose transform (113) are really the same, up to theabove “cosmetic” differences in formalism. In particular, as was shown in [20], our twistorfunction f ( Y ) lives in the adjoint representation of HS symmetry just like the ǫ ( x ; Y ) of[13], while our master field C ( x ; Y ) lives in the twisted adjoint.Now, the authors of [13] proceeded to construct a particular solution to the linearizedbulk equations – the linearized Didenko-Vasiliev black hole – out of a particular covariantlyconstant twistor function ǫ ( x ; Y ): ǫ ( x ; Y ) ∼ e i K ab ( x ) Y a Y b / . (146)Here, K ab ( x ) a generator of the AdS group – specifically, the generator of time translationsin the black hole’s rest frame – normalized as: K ab K bc = δ ac . (147)In the linearized limit, when we consider the “black hole” as a point particle, this is justthe generator of time translations along the particle’s geodesic worldline. Now, consider theembedding space R , of (now, Lorentzian) AdS . There, the particle’s worldline is just theintersection of the AdS hyperboloid with a 2d plane through the origin of R , , spanned bysome simple bivector S µν . We then recognize K ab ∼ γ µνab S µν as the generator of rotations inthis 2d plane. In the embedding-space formalism, this generator is x -independent.All that remains now is to switch signatures to EAdS , with R , embedding space. Theparticle’s worldline becomes a spacelike geodesic, with boundary endpoints ℓ and ℓ ′ , such41hat S µν ∼ ℓ [ µ ℓ ′ ν ] . Imposing the normalization condition (147), we get: K ab = ± γ µνab ℓ µ ℓ ′ ν ℓ · ℓ ′ . (148)We thus see that, upon translation to the present paper’s formalism, the twistor function(146) from [13] is nothing but our twistor function (136) that corresponds to the boundarybilocal! Therefore, its Penrose transform (138), which solves the linearized field equationswith a particle-like source with charges (144), is just the linearized version of the Didenko-Vasiliev black hole from [13]. This justifies our terminology of referring to particles with thepattern of charges (93) as “DV particles”. VII. DISCUSSION
In this paper, we pointed out two new perspectives on the Didenko-Vasiliev “black hole”solution, or rather its linearized version. First, we learned to view this solution in termsof Fronsdal fields generated by an HS-charged particle, with the special pattern of charges(93). We calculated the interaction between such two particles via their HS fields, andfound that the DV pattern of charges has a unique property: in a certain sense, it makes thetwo-particle interaction non-local. Specifically, with this pattern of charges, the interactionaction does not have a short-distance singularity as the two particles are brought closetogether, for almost any angle between the worldlines; the exception is the angle θ = π ,which in Lorentzian corresponds to a particle and antiparticle mutually at rest.Second, we learned to identify the DV solution as the bulk holographic dual of a bound-ary bilocal operator O ( ℓ, ℓ ′ ) = ¯ φ I ( ℓ ′ ) φ I ( ℓ ), and showed that for two such objects, the bulkinteraction action agrees with the connected boundary correlator. In more detail, we sawthat a boundary bilocal operator generates a bulk DV particle that travels along the geodesicbetween the two boundary points, and carries HS charges that source the bulk HS gaugefields; the correlator of two boundary bilocals can then be expressed as the exchange of HSfields between the two bulk DV particles. Though such a picture is new to HS theory, it’sactually been painted before within the general AdS/CFT (or even general CFT) context.We are referring here to geodesic Witten diagrams , introduced in [22, 23] as a bulk rep-resentation for boundary OPE blocks and conformal blocks. Indeed, one way of phrasingthe Flato-Fronsdal theorem is that the bulk HS multiplet (or the boundary multiplet of HS42urrents) is nothing but the OPE of the fundamental boundary fields φ I ( ℓ ) and ¯ φ I ( ℓ ) (withthe identity operator excluded). Thus, our picture of a bulk geodesic stretching between ℓ and ℓ ′ and sourcing the HS multiplet is precisely the “half-geodesic Witten diagram” pictureof OPE blocks proposed in [23] (see figure 1). Similarly, the connected correlator (127) oftwo bilocals can be thought of as a contribution to the 4-point function of the fundamentalboundary fields; our picture of it as an HS field exchange between two geodesics (see figure2) is precisely the geodesic Witten diagram description of conformal blocks proposed in [22].That being said, we view our results as more than just a special case of the geodesicWitten diagram framework. This is because of their particularly tight relation to the basicstructure of HS theory. The bulk DV particle embodies not just some OPE, but the
OPEthat defines the entire spectrum of HS theory. Furthermore, the role of the fundamentalfields φ I , ¯ φ I here is subtle: they carry U ( N ) color, and therefore aren’t usually considered aspart of the CFT’s operators. As a result, the DV particle that embodies their bulk OPE isreally a new ingredient in the bulk theory. It is telling us that, if we wish to accommodateboundary bilocals, then HS gravity must include more than just HS gauge fields interactingwith each other: we must also allow for particle-like HS currents that act as bulk sources forthe HS fields.To make this more concrete, let us point out a specific property that sets the DV particleapart from the HS gauge multiplet. Unlike any of the HS gauge fields, the DV particlecarries electric charge : it is charged under the spin-1 gauge field in the HS multiplet, or,equivalently, under the U (1) part of the boundary color U ( N ). This is easy to understandfrom the boundary perspective: there, the electric charge is carried by φ I , while ¯ φ I carriesan opposite charge. The local HS currents (116), which are the holographic duals of theusual bulk fields, are all electrically neutral, since they contain a product of φ I ( ℓ ) and ¯ φ I ( ℓ ).In contrast, the bilocal operator O ( ℓ, ℓ ′ ) = ¯ φ I ( ℓ ′ ) φ I ( ℓ ) is positively charged at one point,and negatively charged at another; in the bulk, this translates into an electrically chargedDV particle that travels between the two points.Now, suppose that we take seriously the possibility of boundary bilocal operators, andwith them their corresponding bulk HS currents. Then we should ask: what is the generalform of such currents? We’ve seen that a single bilocal insertion produces a particle-likecurrent (43) with the DV pattern of charges (93). What about a general superposition ofsuch insertions? Does the DV pattern of charges for each insertion entail some restrictions43n the resulting bulk currents? It’s easy to see that it does. The relations (93) between thecharges of different spins translate into a linear relation between the corresponding currents,which will be preserved by superpositions. Specifically, the trace of the spin-( s + 2) current T µ ...µ s +2 ends up proportional to the traceless part of the spin- s current T µ ...µ s : T νµ ...µ s ν = − (cid:18) T µ ...µ s − s − g ( µ µ T µ ...µ s ) νν (cid:19) . (149)This relation generalizes the DV pattern of charges (93) to continuous current distributionsin the bulk. It implies that, while the currents at each point x are merely double- traceless,only their fully traceless parts are independent. We conjecture that any configuration ofbulk HS currents that satisfies eq. (149) (and the appropriate conservation laws) arises fromsome superposition of boundary bilocals. For the bulk HS fields, the constraint (149) on thecurrents translates into a relation between the Fronsdal tensors of different spins: F νµ ...µ s ν = 2 s + 1 (cid:18) F µ ...µ s − s − g ( µ µ F µ ...µ s ) νν (cid:19) . (150)Thus, if we allow arbitrary bilocal sources, the spectrum of bulk HS fields gets effectivelyextended, by relaxing the linearized field equations from F µ ...µ s = 0 to (150).What we find exciting is that, even though we dealt here only with linearized HS gravity,the above discussion rhymes with some central issues in the interacting theory:1. In HS theory, which is described via equations of motion rather than an action, itis natural to express the interactions as a coupling between the HS fields and someeffective HS currents, which are in turn non-linear combinations constructed from theHS fields.2. Beginning from the quartic vertex, the interactions of HS gravity suffer from anon-locality problem [37]. In particular, the boundary scalar 4-point function (cid:10) j (0) ( ℓ ) j (0) ( ℓ ) j (0) ( ℓ ) j (0) ( ℓ ) (cid:11) implies a non-local bulk vertex. When this problemwas first glimpsed in [38], it was suggested that the solution may be to include addi-tional degrees of freedom in the description of the theory.3. An attempt to address the locality issue is underway [39–42], with so-called spinlocality replacing ordinary spacetime locality as a guiding principle. As pointed outin [42], this new locality principle is actually ordinary spacetime locality, but with theset of field variables extended to include all possible non-linear currents (but not their44erivatives). Thus, the spin-locality effort is in some sense a concrete realization of thevague proposal in [38] to restore locality by extending the set of degrees of freedom.Now, in the present paper, we also extended the spectrum of bulk HS fields by includingbulk HS currents. We then expressed the 2-point correlator hO ( ℓ , ℓ ′ ) O ( ℓ , ℓ ′ ) i of boundarybilocals as a local bulk process involving both these currents and the original HS fields. Mosttantalizingly, the boundary Feynman diagram for this correlator is very similar to those of theinfamous 4-point correlator (cid:10) j (0) ( ℓ ) j (0) ( ℓ ) j (0) ( ℓ ) j (0) ( ℓ ) (cid:11) , just with two of the propagatorsremoved (see eqs. (127),(137) and figure 2). This leads us to hope that the constructionpresented here may provide a useful alternative viewpoint on the bulk interactions and theirlocality properties. Acknowledgements
We are grateful to Eugene Skvortsov, Per Sundell and Mirian Tsulaia for discussions. Weare especially grateful to Slava Lysov for helping us realize the full extent of the equivalencebetween the DV charges and the cancellation of divergences. This work was supported by theQuantum Gravity and Mathematical & Theoretical Physics Units of the Okinawa Instituteof Science and Technology Graduate University (OIST). YN’s thinking was substantiallyinformed by talks and discussions at the workshop “Higher spin gravity – chaotic, conformaland algebraic aspects” at APCTP in Pohang. [1] M. A. Vasiliev, “Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions,” Int. J. Mod. Phys. D , 763 (1996) [hep-th/9611024].[2] M. A. Vasiliev, “Higher spin gauge theories: Star product and AdS space,” In *Shifman, M.A.(ed.): The many faces of the superworld* 533-610 [hep-th/9910096].[3] J. M. Maldacena, “The Large N limit of superconformal field theories and supergrav-ity,” Int. J. Theor. Phys. , 1113 (1999) [Adv. Theor. Math. Phys. , 231 (1998)]doi:10.1023/A:1026654312961 [hep-th/9711200].[4] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. , 253 (1998)[hep-th/9802150].
5] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large N field theories,string theory and gravity,” Phys. Rept. , 183 (2000) doi:10.1016/S0370-1573(99)00083-6[hep-th/9905111].[6] I. R. Klebanov and A. M. Polyakov, “AdS dual of the critical O(N) vector model,” Phys. Lett.B , 213 (2002) [hep-th/0210114].[7] E. Sezgin and P. Sundell, “Holography in 4D (super) higher spin theories and a testvia cubic scalar couplings,” JHEP , 044 (2005) doi:10.1088/1126-6708/2005/07/044[arXiv:hep-th/0305040 [hep-th]].[8] S. Giombi and X. Yin, “The Higher Spin/Vector Model Duality,” J. Phys. A , 214003 (2013)doi:10.1088/1751-8113/46/21/214003 [arXiv:1208.4036 [hep-th]].[9] G. T. Horowitz and A. Strominger, “Black strings and P-branes,” Nucl. Phys. B , 197-209(1991) doi:10.1016/0550-3213(91)90440-9[10] J. Polchinski, “Dirichlet Branes and Ramond-Ramond charges,” Phys. Rev. Lett. , 4724-4727 (1995) doi:10.1103/PhysRevLett.75.4724 [arXiv:hep-th/9510017 [hep-th]].[11] J. Dai, R. Leigh and J. Polchinski, “New Connections Between String Theories,” Mod. Phys.Lett. A , 2073-2083 (1989) doi:10.1142/S0217732389002331[12] P. Horava, “Background Duality of Open String Models,” Phys. Lett. B , 251-257 (1989)doi:10.1016/0370-2693(89)90209-8[13] V. Didenko and M. Vasiliev, “Static BPS black hole in 4d higher-spin gauge theory,” Phys.Lett. B , 305-315 (2009) doi:10.1016/j.physletb.2009.11.023 [arXiv:0906.3898 [hep-th]].[14] C. Iazeolla and P. Sundell, “Families of exact solutions to Vasiliev’s 4D equations with spher-ical, cylindrical and biaxial symmetry,” JHEP , 084 (2011) doi:10.1007/JHEP12(2011)084[arXiv:1107.1217 [hep-th]].[15] V. Didenko, A. Matveev and M. Vasiliev, “Unfolded Description of AdS(4) Kerr Black Hole,”Phys. Lett. B , 284-293 (2008) doi:10.1016/j.physletb.2008.05.067 [arXiv:0801.2213 [gr-qc]].[16] C. Fronsdal, “Massless Fields with Integer Spin,” Phys. Rev. D , 3624 (1978)doi:10.1103/PhysRevD.18.3624[17] C. Fronsdal, “Singletons and Massless, Integral Spin Fields on de Sitter Space (El-ementary Particles in a Curved Space. 7.,” Phys. Rev. D , 848-856 (1979)doi:10.1103/PhysRevD.20.848
18] S. R. Das and A. Jevicki, “Large N collective fields and holography,” Phys. Rev. D , 044011(2003) doi:10.1103/PhysRevD.68.044011 [hep-th/0304093].[19] M. R. Douglas, L. Mazzucato and S. S. Razamat, “Holographic dual of free field theory,”Phys. Rev. D , 071701 (2011) doi:10.1103/PhysRevD.83.071701 [arXiv:1011.4926 [hep-th]].[20] Y. Neiman, “The holographic dual of the Penrose transform,” JHEP , 100 (2018)doi:10.1007/JHEP01(2018)100 [arXiv:1709.08050 [hep-th]].[21] A. David and Y. Neiman, “Higher-spin symmetry vs. boundary locality, and a rehabilitationof dS/CFT,” [arXiv:2006.15813 [hep-th]].[22] E. Hijano, P. Kraus, E. Perlmutter and R. Snively, “Witten Diagrams Revisited: TheAdS Geometry of Conformal Blocks,” JHEP , 146 (2016) doi:10.1007/JHEP01(2016)146[arXiv:1508.00501 [hep-th]].[23] B. Carneiro da Cunha and M. Guica, “Exploring the BTZ bulk with boundary conformalblocks,” [arXiv:1604.07383 [hep-th]].[24] H. Y. Chen, L. C. Chen, N. Kobayashi and T. Nishioka, “The gravity dual of Lorentzian OPEblocks,” JHEP , 139 (2020) doi:10.1007/JHEP04(2020)139 [arXiv:1912.04105 [hep-th]].[25] Y. Neiman, “Twistors and antipodes in de Sitter space,” Phys. Rev. D , no. 6, 063521(2014) [arXiv:1312.7842 [hep-th]].[26] T. Curtright, “Massless Field Supermultiplets With Arbitrary Spin,” Phys. Lett. B , 219-224 (1979) doi:10.1016/0370-2693(79)90583-5[27] I. L. Buchbinder, A. Pashnev and M. Tsulaia, “Lagrangian formulation of the masslesshigher integer spin fields in the AdS background,” Phys. Lett. B , 338-346 (2001)doi:10.1016/S0370-2693(01)01268-0 [arXiv:hep-th/0109067 [hep-th]].[28] A. Mikhailov, “Notes on higher spin symmetries,” [arXiv:hep-th/0201019 [hep-th]].[29] E. Joung and J. Mourad, “Boundary action of free AdS higher-spin gauge fields andthe holographic correspondence,” JHEP , 161 (2012) doi:10.1007/JHEP06(2012)161[arXiv:1112.5620 [hep-th]].[30] A. Campoleoni, M. Henneaux, S. Hrtner and A. Leonard, “Higher-spin charges in Hamiltonianform. I. Bose fields,” JHEP , 146 (2016) doi:10.1007/JHEP10(2016)146 [arXiv:1608.04663[hep-th]].[31] Y. Neiman, “Antipodally symmetric gauge fields and higher-spin gravity in de Sitter space,”JHEP , 153 (2014) [arXiv:1406.3291 [hep-th]].
32] I. F. Halpern and Y. Neiman, “Holography and quantum states in elliptic de Sitter space,”JHEP , 057 (2015) doi:10.1007/JHEP12(2015)057 [arXiv:1509.05890 [hep-th]].[33] Y. Neiman, “Higher-spin gravity as a theory on a fixed (anti) de Sitter background,” JHEP , 144 (2015) doi:10.1007/JHEP04(2015)144 [arXiv:1502.06685 [hep-th]].[34] R. Penrose and W. Rindler, “Spinors And Space-time. Vol. 2: Spinor And Twistor MethodsIn Space-time Geometry,” Cambridge, Uk: Univ. Pr. (1986) 501p[35] R. S. Ward and R. O. Wells, “Twistor geometry and field theory,” Cambridge, UK: Univ. Pr.(1990) 520p[36] M. Flato and C. Fronsdal, “One Massless Particle Equals Two Dirac Singletons: ElementaryParticles in a Curved Space. 6.,” Lett. Math. Phys. , 421-426 (1978) doi:10.1007/BF00400170[37] C. Sleight and M. Taronna, “Higher-Spin Gauge Theories and Bulk Locality,” Phys. Rev. Lett. , no. 17, 171604 (2018) doi:10.1103/PhysRevLett.121.171604 [arXiv:1704.07859 [hep-th]].[38] A. Fotopoulos and M. Tsulaia, “On the Tensionless Limit of String theory, Off - ShellHigher Spin Interaction Vertices and BCFW Recursion Relations,” JHEP , 086 (2010)doi:10.1007/JHEP11(2010)086 [arXiv:1009.0727 [hep-th]].[39] O. A. Gelfond and M. A. Vasiliev, “Homotopy Operators and Locality Theorems inHigher-Spin Equations,” Phys. Lett. B , 180 (2018) doi:10.1016/j.physletb.2018.09.038[arXiv:1805.11941 [hep-th]].[40] V. E. Didenko, O. A. Gelfond, A. V. Korybut and M. A. Vasiliev, “Homotopy Propertiesand Lower-Order Vertices in Higher-Spin Equations,” J. Phys. A , no. 46, 465202 (2018)doi:10.1088/1751-8121/aae5e1 [arXiv:1807.00001 [hep-th]].[41] V. E. Didenko, O. A. Gelfond, A. V. Korybut and M. A. Vasiliev, “LimitingShifted Homotopy in Higher-Spin Theory and Spin-Locality,” JHEP , 086 (2019)doi:10.1007/JHEP12(2019)086 [arXiv:1909.04876 [hep-th]].[42] O. A. Gelfond and M. A. Vasiliev, “Spin-Locality of Higher-Spin Theories and Star-ProductFunctional Classes,” JHEP , 002 (2020) doi:10.1007/JHEP03(2020)002 [arXiv:1910.00487[hep-th]]., 002 (2020) doi:10.1007/JHEP03(2020)002 [arXiv:1910.00487[hep-th]].