JHEP10(2020)127 Springer July 4, 2020 : October 20, 2020 : September 22, 2020 : boundary action. Received Published Accepted spin-local Published for SISSA by https://doi.org/10.1007/JHEP10(2020)127 disagrees with the usual CFT partition HS Z [email protected] , . 3 2006.15813 The Authors. Higher Spin Symmetry, AdS-CFT Correspondence, Discrete Symmetries, c

We consider the holographic duality between 4d type-A higher-spin gravity and , [email protected] seriously. We are rewarded by the dissolution of a long-standing pathology in HS Z can be recovered from first principles, by demanding a E-mail: Okinawa Institute of Science1919-1 and Tancha, Technology, Onna-son, Okinawa 904-0495, Japan HS Open Access Article funded by SCOAP ArXiv ePrint: Z Keywords: Higher Spin Gravity Feynman diagrams with different permutations of theare source instead points. ambiguous, In spoiling Euclidean,a the would-be signs conflict linear between superpositions. boundary localitytake Framing and the HS situation symmetry, we as higher-spin sacrifice dS/CFT. locality and Though choose we to lose the connection to the local CFT, the precise form of be encapsulated in an HS-algebraicnot twistorial expression. just on This separate expression canso boundary be for insertions, evaluated the but first onfunction. time, entire finite and source While find distributions. that sucheven the We disagreement in result do their was absence. expected Wecalculations. ascribe due In it to to Lorentzian a boundary contact confusion signature, corrections, between this on-shell it manifests and via persists off-shell wrong boundary relative signs for Abstract: a 3d free vector model. It is known that the Feynman diagrams for boundary correlators can Adrian David and Yasha Neiman Higher-spin symmetry vs. boundaryrehabilitation locality, and of a dS/CFT JHEP10(2020)127 3 2 26 33 6 27 7 27 30 12 33 bases 11 3 R basis 3 17 R 22 22 1 31 16 – i – 23 13 21 27 modes 8 3 15 R modes 34 7 14 3 S 11 7 23 35 7 1 field 3.3.2 Real contour and relation to the 3.2.1 Bulk spinor-helicity basis and its interpretation as3.2.2 a boundary master Boundary3.2.3 momentum basis and its Importance star and products disadvantages of the 3.3.1 Basis, star products and partition function 2.3.4 Fixing the sign factors 2.2.3 The Penrose transform 2.3.1 The2.3.2 CFT picture The2.3.3 twistor picture: spin From 0 spin 0 to bilocals to all spins 2.1.2 Twistor2.1.3 space Local spinor spaces 2.2.1 HS2.2.2 algebra Star products with spinor delta functions 2.1.1 Spacetime 4.2 Complex conjugation 4.3 Spin parity 3.4 Disagreement with the CFT result 4.1 Spontaneously symmetry breaking as the culprit 3.3 Bulk master field, 3.1 From correlators3.2 to partition function Boundary master field, 2.3 Boundary correlators 2.2 Higher-spin algebra and the Penrose transform 1.3 A boundary1.4 puzzle: CFT Embedding correlators, space partition and function spacetime-independent and twistors HS symmetry 2.1 Geometry 1.1 Summary and1.2 structure of the The paper bulk situation: HS gravity, locality and spin-locality 4 Explanation: sign ambiguity and discrete symmetries 3 Conflict: Euclidean partition functions 2 Euclidean boundary correlators from HS algebra Contents 1 Introduction JHEP10(2020)127 44 ]. 1 4) symmetry, , 49 48 39 51 42 43 36 locality is perfectly compatible with HS 45 41 36 37 spin- 41 – 1 – 46 3) symmetry, a subtle sign disagreement appears , 40 49 45 HS Z 7.2.1 A7.2.2 global maximum for the A Hartle-Hawking road wavefunction towards physics on cosmological horizons 7.1.1 Leaving7.1.2 the local CFT behind A7.1.3 spin-local path integral Integer spins, antipodal symmetry and reality conditions 6.2.1 Twistor6.2.2 space as phase space Boundary spinor space as configuration space The conflict that we find between local and HS-symmetric partition functions appears 7.1 Boundary spin-locality 7.2 Implications for dS/CFT 6.3 On-shell HS algebra vs. off-shell CFT correlators 5.1 Boundary topology5.2 and propagators Twistors, spinors and star products 6.1 Dimensionality of6.2 algebras HS algebra as quantum mechanics for both Euclidean and LorentzianFor boundary a signatures; Lorentzian however, the boundary,already details with are at SO(2 different. the levelthe of correlators correlators. can beintegrating For them made a into to partition Euclidean agree, functions. boundary, but In with then the SO(1 interest a of subtle painting failure a full of picture, linearity we occurs will when symmetry leaves unconstrained. Choosing theover the spin-local, standard HS-symmetric local partition function one,at we finite arrive at sources, closed-form decomposed expressionsfunction for is in the free SO(4) partition of function modes.Hawking a wavefunction long-standing for Aside pathology a from in constant spin-0 higher-spin its source dS/CFT, simplicity, was wherein this not the globally partition Hartle- peaked at zero [ spin (HS) holography, with thehorizon in long-term dS/CFT. goal During of these extractingboundary explorations, locality physics we and came inside to global a notice HSliterature. cosmological that symmetry the is conflict sharper On between than the wassymmetry, apparently and other expected can in hand, even the boundary be invoked to fix those details of the partition function that HS 1 Introduction 1.1 Summary and structureThis paper of arose from the the paper study of non-local expressions for the partition function in higher- 7 Aftermath: choosing 6 Explanation 2: on-shell vs. off-shell boundary particles 5 Conflict 2: Lorentzian boundary correlators JHEP10(2020)127 , , , , x Y 7 7.2 with the disagree . Moreover, to fully define Y boundary data translates into space, which until recently has Z and 4) case, which is the relevant one Z , x of independent derivatives). The second , after introducing the necessary geometry 2 , we discuss how boundary spin-locality can , we switch to a Lorentzian boundary. There, )-dependent fields. -point correlators in HS-algebraic language. In – 2 – 5 7.1 n , we upgrade the correlators into an HS-algebraic x, Y 3 ). In section , is entirely auxiliary. Thus, the Vasiliev equations are not ]. However, higher-spin gravity is a theory of infinitely 7.2 6 full infinite tower Z , we explain the disagreement in terms of sign ambiguities , we provide another explanation for the disagreements, in 4 6 ]. In section ] is a most fascinating and frustrating specimen of mathematical 3 , 5 2 , 4 , we actually get the Y -dependence, to get the actual field equations in In a more standard field theory, the freedom of field redefinitions is crucially curtailed Like string theory, HS gravity a theory of infinitely many fields, including a massless The paper is structured as follows. In the remainder of this section, we review some Z not been done ina a freedom principled of manner. redefining Ambiguity the in “physical” the ( by the restrictionmany of massless locality fields, [ which interact at all orders in derivatives. Therefore, the task of component fields in the HSture multiplet, of along with theirtwistor-like variable, derivatives known (due as to thequite bosonic field na- equations, butthe rather “equations for thethe field system, equations”: one must one decide must on first boundary solve data in the spin-2 “graviton”. Inery its even) simplest spin, version, allgiven there massless. instead is by a The Vasiliev’s single theorythese equations field equations does of of involve not motion. two every haveis twistor-like In integer a pairs analogous addition (or known of to to ev- action spinor the spacetime principle, variables. coordinates fermionic but The coordinates is first, in known a as superspace formalism: it enumerates the physics. It can beof thought spacetime of symmetry, as analogous an toencounters exploration in the of supergravity. finite-dimensional infinite-dimensional, While fermionic bosonic thewill extensions extensions theory be that can interested one be here formulated in in the various 4d dimensions, case. we replace the standard structurewe of discuss the the CFT consequences as for a dS/CFT. guiding1.2 principle. Finally, in The section bulkHigher-spin situation: gravity HS [ gravity, locality and spin-locality the signs in HS-algebraicCFT correlators are calculation. no In longerterms section ambiguous, of but a they confusionwe between discuss on-shell the and implications off-shell ofby boundary choosing the particle the local states. HS-algebraic partition CFT. In function section over Specifically, that in given section the process, we presentspacetime-independent for twistor the functions, first andin time fix our the some previous dictionary work crucial betweenformula [ sign local for mistakes/ambiguities CFT the sources EuclideanCFT and calculation. partition function, In andin section the show HS-algebraic that correlators. it In disagrees section with the local for dS/CFT. relevant background and recent developments indS-specific HS gravity discussion and to its section holography (leavingand the algebra, we construct the Euclidean discuss both cases; however, our main focus is the SO(1 JHEP10(2020)127 ] ], 20 20 , 19 ], centered (note that 12 ]. In [ Y – 9 18 ], these operators ]. In the simplest 17 16 ]). The authors of [ 21 , including an “honorary current” of ]. Either way, the apparent failure of ) (for all integer spins in the bulk), or s 22 N vector model of spin-0 fields. This will be our – 3 – free ] is one of the main motivations behind the ongoing 20 directly, but in the spinor coordinates ]. This is now a subject of intense work [ x 8 -point functions of the boundary theory [ , one for each spin , -space are related by the Vasiliev equations). So far, it s n 7 Y ...k ) 1 s ( k J ] actually reached the opposite conclusion, and was corrected 19 space, and thus succeeds in constraining the theory and completing its ] of spin-locality as an alternative guiding principle. Z -space and in ]. Generally speaking, higher-spin theory is holographically dual to vector 12 x – contradiction of this general nature was long anticipated in the literature. 15 9 – 13 some ], the boundary theory is just a ]; for an early hint of the quartic vertex’s non-locality, see [ 17 ) (for even spins only). In this paper, we will enact a similar drama in miniature, this time on the boundary If one assumes the holographic duality, one can bypass some of the difficulties with the These recent developments were sparked in part by holographic computations. Simi- 20 N ( When speaking of thelevels boundary of CFT discussion. and itsprimary In locality, operators AdS/CFT, one the and must bulkare distinguish their fields all between sources. conserved are two currents mapped In to the the free CFT’s vector single-trace model of [ out a contradiction betweenreview, its local structure and global HS symmetry. As we will now 1.3 A boundary puzzle: CFT correlators, partition function and HS symmetry reaching parts of this claimspacetime locality have that been was challenged noted [ exploration in [ [ side of higher-spin holography. While the boundary CFT is necessarily local, we will point vertex remains non-local atoriginal all calculation distances, much in likein [ a [ massless propagator (notewere that thus the led toin claim the that bulk locality theory, can rendering in the no latter’s way interactions be completely used arbitrary. to The constrain most field far- redefinitions bulk equations. In particular,fields, one starting can reconstruct from the the interactionthis known vertices was of physical done bulk fortheory’s non-locality the may quartic be scalar restricted vertex. to the The cosmological result radius. negated earlier Instead, hopes the that quartic the dualities [ models, with acase Chern-Simons [ gauging ofcase the of interest, internal since symmetrymodel’s it internal [ manifests symmetry group HS canO symmetry be in either the U( most direct manner. The vector seems that this criterionin is the non-trivially auxiliary fulfilled bydefinition. a particular choice of boundary data larly to string theory, higher-spin gravity appears on the bulk side of AdS/CFT holographic locality involved. We wantenough a to notion constrain the of theory locality [ around that the is criterion loose of enoughnot to “spin-locality”. in be the true, This spacetimederivatives but concept coordinates in tight refers to constraining non-locality properly defining the theory becomes tightly intertwined with understanding the non- JHEP10(2020)127 . . ). s s n ]): ...k ...k ) to a ; for ) 2 (1.1) 1 1 s s ( σ ( k k -point ] (and J n (2) A ij 25 , . . . , ` J , familiar ). For the 1 I 1 8 ` n . φ I I = to a Maxwell ¯ φ φ i I ij A ! φ T , . . . , ` i i to a source 1 ∂ A ↔ ` I I φ ¯ φ from the one in [ I i traces ¯ s φ − ) can fail: the integrals gauge potentials − n . For example, for spin 0, ) = s = s s k ...k (0) ∂ ) → (1) i 1 s J ( k J , . . . , ` ]). At the end of the day, even A ... 1 is the vector model’s fundamental = ` ]). 27 distinct points ( i I +1 , J φ m n k 29 26 , ∂ → m 28 k , ∂ 1 ← to a curved background metric. Analogously, of the stress-energy tensor I ... ij -point correlators are then given by derivatives φ – 4 – 1 k n T ( -point correlation functions of the currents ij ∂ n ← h  1 2 of the scalar operator s m I -point correlators over the insertion points ( 2 2 φ n I  of the charge current ¯ to the external potentials φ m i σ s J 1) i ...k ) − 1 A s ( k ( J =0 s , and so on. The s ). Naively, this can be written as a Taylor series, whose coefficients . The sources for these operators are spin- X m ` ij I ( ...k h )

s 1 s ( k I ¯ ...k φ ) A 1 s s ( k ) are boundary spacetime indices, and 1 i A 2, the contact corrections should represent a higher-spin generalization of ] (note that the definition here differs by a factor of 2 = ; for spin 2, a coupling i s 24 A , ) are all conserved. In particular, it should be possible to package all the s > ...k ) 1 s 23 ( k i, j, k, . . . 1.1 J Can we do better? While individual higher spins may be difficult, we might expect some At a more advanced level, one can ask for the partition function at some nonzero value simplification once the entireSpecifically, HS the multiplet free is vectorboundary treated model conditions) in (or, should a equivalently, enjoy the global unified,rents bulk higher-spin ( HS-covariant HS symmetry, manner. since theory the with higher-spincorrelators appropriate cur- in an HS-covariant manner. This has been carried out several times, in slightly unknown, though they caneven all uplifted be to derived formin in interacting principle the bulk from HS free the theories vectorpartition construction [ function model, of remains in a [ difficult whichbackgrounds open are all problem of the (though course calculations correlators possible for — can some see low-spin be e.g. written [ out explicitly, the corrections are absent; for spinfrom 1, scalar we have electrodynamics; the for singlewhich comprises contact spin the correction 2, non-linear coupling there of for is spin an infinitecoupling tower of to contact a corrections, curved background. The explicit form of these corrections remains largely here. Even perturbatively,are the typically naive UV-divergent! integrationto over The ( the solution sources is inpoint to are the multiplied include Lagrangian, together. additional, i.e.these For “contact terms non-linear the corrections” in couplings special can which case also two in be or which deduced the more from sources sources gauge are invariance. at gauge the For fields, spin same 0, such don’t involve any integrals. of the sources are just found by integratingThe the Taylor series might miss some non-perturbative effects, but those will not bother us potential metric perturbation at zero of the pathfree integral with vector respect model, to all sourcesgiven such at by correlators 1-loop can Feynman diagrams, be which written (in down spacetime, without as calculation: opposed to they momentum are space) The most basic objects of studyThese are can the be extracted froma the linear CFT coupling path integral,we in add which one a adds mass-type to coupling spin the 1, Lagrangian we add a coupling Here, ( field with color index spin 0 [ JHEP10(2020)127 n ]. A ], em- 33 of this , 35 2 32 with some ` . (0) ]. There, all the J 31 ]. In section 36 in the free vector model ], based on two pieces of 2 ) s ( ] were careful to warn that J in an HS-covariant form. In 30 (0) -point correlators and obtain the J n ], then streamlined in [ . 30 Y – 5 – their coefficients were derived from the boundary ; thus, in the presence of sources, we should expect with ], using a boundary HS-covariant formalism from [ correlators! Also, the currents 34 ) for the spin-0 operator ` ij ( T σ as a function of finite sources, and these are manifestly conserved. s away from sources ...k ) 1 s ( k ], the correlators J 2 all other spins. practice, this means replacing theappropriate dependence spinor on or spacetime twistor coordinates variables ], the Vasiliev equations can be written equally well on a non-dynamical, pure (A)dS By HS symmetry, this expression can now be extended to provide the correlators for Encode the source Find an HS-covariant expression that reproduces the correlators of This is all very confusing! Can we naively integrate the correlators to make a sensible On the other hand, we know that for spin 0, contact corrections are absent. Thus, 40 3. 1. 2. geometry in the bulk.in [ However, evengeometry. this may not be cause forpartition panic: function, as or not? pointed out The usual local treatment of the CFT says “no”. Global HS evidence. First, the HS-covariantnot twistorial show expression any that sign offor one a the obtains currents UV in divergence. thisThus, Second, way from does one the can point derive ofcontact from view corrections it of appears expectation gauge to invariance values above, and be it also seems gone! of to UV do This divergences, away with is the curved need a backgrounds for strange on the conclusion, boundary, since, and with as dynamical discussed contact corrections to restore (gauged) higher-spin symmetry. in that sector, wepartition should function. be able to Butpartition just function then, integrate for by all the global spins? This HS was symmetry, proposed shouldn’t explicitly in this [ also give the full the correlators are just the“meat” “trivial”, is kinematic in part of theof the contact curved theory, corrections. and geometry, that and Indeed, thecontact thus dynamical for corrections the spin to gravitational the 2, natureare of in only the some conserved bulk sense theory, the arises entire from notion the So far, no controversy.to Confusion the begins partition when function one at attempts finite to sources. upgrade from The correlators authors of [ similar answer was given inbedded [ in the largerpaper, framework we of will so-called again multiparticle derive algebralanguage, these [ and correlators, working from up there from tobe the all spin-0 distilled spins. case as: The to minimal the version bilocal of all the above procedures can relators from HS invariants wascorrelators proposed were in packaged [ HS-covariantly,(which, and assuming fixed the up existence of tostructures). a a boundary In CFT, single [ can coefficient intheory turn for in be each an fixed HS-algebraic by manner, matching OPE starting from a boundary-bilocal formalism [ different versions. From a bulk point of view, the general philosophy of constructing cor- JHEP10(2020)127 of contact ]. Namely, we define 2 , both in general and with 7 ] will provide us with a con- not a consequence ]; the main difference is that ]. This is in contrast to most 40 , 38 40 , , 39 37 , 2 39 , 2 – 6 – ]. Contact corrections will be absent, and the notions of a twistor argument, which makes no reference, implicit or 35 , 34 just The spacetime-independent twistor language of [ This is all standard in Penrose’s twistor theory [ After carefully analyzing the disagreements and their causes, we will choose one parti- In this paper, we claim to resolve the above confusion. In Euclidean boundary signa- spacetime, but to a pair of local spinors at avenient, particular covariant spacetime middle point. groundpoints. between These the “reference “reference frames”the frames” standard themselves, (bulk of which and we different boundary) will spacetime master-field also formalisms use from extensively, the are HS just literature. Penrose would usually consider conformalmological constant, symmetry which in only 4d, leaves isometriessymmetry whereas of on we the the fix 4d 3d a bulk boundary).transform (or, nonzero Within equivalently, were cos- conformal HS introduced theory, Penrose-style by twistorsof one and the the of HS Penrose the literature, authors where [ the word “twistor” refers not to a global geometric object in spinor arguments. In thisconsideration. case, there There is, remains however, a antransform, dependence alternative: as on the a the field function spacetime can of explicit, point be to under encoded, any via spacetime the point. Penrose the 4d bulk andas 3d the boundary in spinors terms ofpoint. of this Specific a embedding bulk 5d space, or embeddingbe with boundary space. viewed no points as We are need subspaces then associated (or to definebe with quotient reference twistors written spaces) local with of any spinor spinor twistor bulk spaces, indices, space. or which or Bulk packaged can boundary and into generating boundary functions fields (“master can fields”) with 1.4 Embedding space and spacetime-independent twistors In this paper, we will use spinors and twistors after the fashion of [ locality, replacing the latter by spin-locality.one The already resulting put formalism forward will in becurved essentially [ background the and a dynamical bulkthis geometry outcome indeed may seem seem, to we beregard will lost. to argue the Radical for though project its of merits dS/CFT. in section either case, we have a conflictingly, between the boundary disagreements locality and begin global alreadycorrections. HS at symmetry. spin Surpris- 0, so they are tion function over the other. We will choose to retain HS symmetry at the cost of boundary on their integrals? Is linearity somehow failing? ture, there is indeedand a twistor failure languages. ofbetween linearity For the Lorentzian hidden CFT boundary, inside path there the integral is dictionary and instead between HS algebra the a already local subtle at disagreement the level of the correlators. In symmetry says “yes”. If the two formalisms agree on the correlators, how can they disagree JHEP10(2020)127 : 4 µ , ). ), = 4 ` → 1 , 2). 1 ` , R µ 2.1 , i.e. · (2.2) (2.1) R 4 , ` : ∈ d` l 1 vectors tangent µ is a unit R µ, ν, . . . µ ∞ 4 d` ` 4 , with 1 with , normalized µ R µ ` ` ) bulk with an l is just the flat 4 EAdS ∈ . 4 µ ” via the limiting µ 0 ` , where x α` 1 2 EAdS − that are tangential to + EAdS µ . 4 , = v Minkowski space, with a 1 on

4 R , ∼ = 0 ∞ µ 1 1, where ` ∈ µ · R > ∇ − v lightcone. An especially useful µ ` 0 v 4 = , bulk. In this frame, the boundary 1 0 , x 4 x R 1 · − . ` µ = EAdS zl µ . x 2 that are tangential to the hyperboloid ( + ` µ along the geodesic connecting · 4 x µ , . The double cover of this group is USp(2 | ` 1 x 1 . 4 ` 4 , – 7 – 1 z ν , R 2 1 1 x is the 3-sphere of lightlike directions in R − = ∈ µ R 4 x µ µ √ ∈ v x + = µ ) is said to have conformal weight ∆. Boundary ν µ x µ r δ acting as the “point at infinity”. The distance between ` EAdS  ( is . Such rescalings induce Weyl transformations , and locate goes to zero. Sometimes, it is important to also specify the f = ∞ µ , or the conformal group of its boundary, is just the rotation 1 2 3 ) = vector, i.e. a boundary point. In this frame, the boundary ` = 0. The covariant derivative ∆ z 4 4 , which is used to raise and lower spacetime indices ( x R ρ` − − µ ( can be said to “approach the boundary point ρ ν µν µ x = q η x µ → l v · EAdS EAdS µ ) = = 0, modulo the equivalence relation , with ` ` µ 3 ≡ , where . Another useful frame is defined by ` lightlike 3 , followed by projecting all tensor indices back into the R · ρ` x 4 /z as S , ( are simply vectors v · 1 µ f in this flat µ 4 ` v R ` 2 → in , ` 4) in the embedding space can be described as embedding-space vectors 1 µ , µ ` ` x EAdS ∂ on the boundary’s conformally flat metric. A boundary quantity that depends µ ]. d` 2 µ Finally, a bulk point A conformal frame on the boundary corresponds to choosing a particular vector The conformal boundary of boundary. We will represent these as embedded in a flat d` 3 2 2.1.2 Twistor space The isometry group of group SO(1 procedure direction of the approach. Inwith that respect case, to we choose a second boundary point conformal frame is definedfuture-pointing by timelike the vector, lightcone i.e. a section pointis in a the unit 3-sphere is a future-pointing becomes flat space two points on the scaling as at a point the lightcone, i.e. along each null direction, i.e. choosing a section of the space with the projector the projective lightcone. We represent0, boundary modulo points rescalings by lightlikeρ vectors Vectors in i.e. that satisfy derivative We consider first Euclidean signature, i.e.S a Euclidean anti-de Sitter ( (mostly-plus) flat metric The 4d bulk is defined as the hyperboloid of unit future-pointing timelike vectors in 2.1 Geometry In this subsection, we introduce3 our of geometric formalism, [ in a condensed version of section 2.1.1 Spacetime 2 Euclidean boundary correlators from HS algebra JHEP10(2020)127 , 4 ] C cd I (2.5) (2.6) (2.3) (2.4) ab [ to map I ), where µ σ and under γ i = 3 . component- . We define − c ab , ab I are real under V f abcd , which satisfy c  bivectors: , b b b = a µν ) ab a 4 that is consistent 4 ) ∗ , ) ν γ ) 1 A µ µ γ a 1 4 γ ν γ ab R ¯ tr U ` , I ( = 1 4 subspaces out of the b ). Twistor space has a a → 2 (0) ) and µν − f µ a C f 2 γ µν
U µ 2 γ . ; x ) = a, b, . . . A ( d a U µν ( tr U f f dU are symmetric. We use ) c 1 8 ≡ ab 2 blocks made up of (1 µν U γ ( µν dU factors for convenience): = × 1 2 γ b π A U δ = dU 4 Ux`V a d , which we use to raise and lower indices as ab b c f δ Z subspaces of twistor space (though these are dU . The gamma matrices ( 4) spacetime symmetry. As twistor matrices, , – 8 – ; det 2 ab 2 = ) ; I , but cannot all be real component-wise. Instead, ; C π A µ ac ab . We also introduce the antisymmetrized products abcd γ = I 1 1. Therefore, it can’t be represented as ξ  are the Pauli matrices. iV U µν ab  4!(2 − det = µ ab ab η I σ vectors, and similarly for ¯ γ I , we can construct a Levi-Civita symbol µ √ 2 V e ≡ 4 1 4 γ , , and there is no invariant notion of a real twistor. The 4 ab − 1 ∗ = I d − U ) R 4 2 a = / d = Z ) U } (in which we include 2 µ ν ξ , since he prefers to work projectively). To see how this works, ) = U 1 , γ UAU . From ( 4 µ U ) will be important below. ; ) on twistor space via the formal integral: d with inverse can be defined as real, both component-wise ( ba ( γ CP δ I { U µ U e b ( 2.6 ab ab ξ spacetime points as 4 δ I I U d ab µ γ = . We will denote twistors by the indices ( Z ], which generate the SO(1 a = 4) signature, the properties of twistors under complex conjugation follow ν , U 2 identity matrix, and , γ ab 4) symmetry squares to defines , is a symmetric twistor matrix. Sign ambiguities of complex Gaussian integrals , ξ µ representation, composed of 4-component Dirac spinors, is what we will refer to b × γ [ U 1 2 ab 1 2 are traceless and antisymmetric, while the ab A I complex conjugation ( µ ≡ We will often use condensed index-free notation, in which twistor indices are implicitly γ = twistor space a µν 2.1.3 Local spinorChoices spaces of (bulk or boundary) spacetimetwistor points space. pick out These various areessentially the local spinor spacesusually at described each spacetime as point. In fact, Penrose where such as the one in ( the delta function and note the formula for Gaussian integrals: the twistor complex conjugation ¯ they are quaternionic, i.e. they1 can is be the 2 written in 2 contracted bottom-to-top. Thus, for example, those of SO(3) spinors.with SO(1 The complex conjugationwise operation symplectic metric the twistor complex conjugation, between twistor matrices and In our SO(1 Twistor space andthe spacetime Clifford are algebra relatedγ by the gammathe matrices ( as symplectic metric U along with a measure Its spin- JHEP10(2020)127 , ) 4 ξ − ) for ( (2.7) (2.8) (2.9) ξ (2.10) (2.11) (2.13) P ( EAdS over the P ξ ) and , which gives ), i.e. on the . The rank-2 ξ µ ξ ( x ; (2.12) , x . − P 0 2 ( − ξ ) P · = (0) A can be decomposed subspace associated ) ξ breaks the spacetime f µ 1. In Lorentzian bulk 2 U ξ 0 U ξ x ) ( C ξ − ξ ) + P · 0 − = ξ ξ ( · tr ( µ . Accordingly, twistor space P 0 − x ) x 1 2 ξ 0 . The , ) µ 0 4 ξ , via: , )( − x ξ ( )  1 . . 0 ξ · b ξ . The subspaces 2 P · R ) 0 ( a . a U 2 ξ ξ ξ ) ξ u ) = ( ) ξ ∈ ( U 0 ( u )( A , we must use ξ + + 2 µ ( ξ ( iv p x ) ξ ξ d a b · e u ξ ) 2 ) ( ξ itself is a point on the antipodal = ξ ξ d ξ δ ( u ( ( ) ) · µ v 0 ξ p ξ 2 ab x = ( ξ ( 1. The choice of point d : ; det u 0 P − u − ) 2 − U ξ ξ ) d ( – 9 – ) = ≡ p ) = A P  0  ) ( 0 ξ Z ξ x ξ ( ) as ξ 1 b ( ; 1 2 · 0 · u ξ  x 0 ( ( ξ du π det ξ ) = f ) 2 · ) ξ ) = rather than simply ) U a ( p ,P 0 ξ ξ ( ) ) ) + ξ ( with U ξ 0 ξ ( ξ 2 is a projector, which spans (or projects onto) the right- b du a ( ) 2. Note that = δ ξ ( 0 a u µ / u / · ) reads: ( ξ ) 2 ) + x ) P ) can be defined as: 0 P 0 ξ / ξ x ξ ) x ξ − δ ( ξ ): ξ ( in index-free notation. We will also use the notation ( = · ( ) ξ 0 P )( − 0 ( P
P ξ µ ξ ξ ( ) Au ξ ξ P · ) ξ u ξ )( ( ( 2 on ξ + ξ u ( d ) P · ξ e ) ξ 2 ) = (1 0 ) = (1 + ( · ) ξ p ξ ξ is spanned by the rank-2 twistor matrix: x ( x u (  ξ ( ( 2 4) down to the bulk rotation group SO(4) at P − µ u p 1 2 − d , ( P 2 Z ξ d √ P = = ) ξ ) ( 1 2 is again a symmetric twistor matrix. Finally, a twistor ξ U ( P 4 u Z d ab subspace itself. When we want to emphasize that a twistor belongs to the subspace A ) = ξ 2 ). To describe the left-handed spinor space at ( Let’s now consider more concretely the local spinor spaces at bulk vs. boundary points. A measure x C ), we will denote it as e.g. ( P ξ ( ab twistor matrix handed spinor space. TheP restriction of the twistor metricthe onto projector this spinor spacei.e. is on simply the other branch of the double-sheeted hyperboloid At a bulk point, we take symmetry SO(1 decomposes into the right-handed and left-handed Weyl spinor spaces at where along a pair of spinor spaces The Gaussian integral on A useful integral is thatspinor of space a associated delta with function another associated point with one spacetime point With this measure,subspace we orthogonal define to a delta function, whose support is on P are totally orthogonal under the twistor metric. with the point or the consider a spacetime point represented by a vector JHEP10(2020)127 ) ) ` ( ; in ∗ ` 2.13 P (2.19) (2.18) (2.14) (2.15) (2.16) at ) in the subspace 2.8 2 C between are represented ) ` ` cospinors ( = 0. This breaks u ` at · → preferred ` ) ` ∗ ( u , . one ) ]. The measure ( ) ` with Spinors . b ( ` ): ; (2.17) ) ( ` 41 ) µ . l , ( b x ` du · µ P ) ) µ  ` P ) ` ` of a twistor into right-handed and b ( ` a ( ) is the space of ∈ ( = ∗ ( ∇ ) ` , and every spinor index back into ) π x ( ). It will be convenient to define a du µ du ` cd µ ν 4 ) du a ( x − γ ξ P , and the integration measure can be x ( x ( . ) ab a µ l  u , there’s no longer a need to introduce ab  x ) ) a ( l x ` ) ` ( , which is totally null under the twistor − + ), with the measure decomposition ( ( ` b , u 1 2 ∗ ( ` ) (1 du d u ) + x 1 2 1 2 ` 2 = ) ( a ( P du and ν µ d ) ) x u ) 2.12 ( u ) l δ ) = ∗ ( ` ` x π ) ∗ (  ) ` u ≡ + ` ( ( – 10 – ) = ) = − u point, i.e. π 2 ( , followed by projecting every tensor index back ) ( ` can now be extended to objects with Weyl spinor 2 x ab d 2(2 ` ab ( U µ a ) = d a ( ab ) c µ l P  P ∂ ) x 2(2 P P ( · − x ∗ ( 2 P ∇ ν µ ` u = P − ( ≡ = q ( ) ≡ − ) . ) become canonical representatives for the equivalence x P ` l ) ∼ = defines a natural isomorphism U ∗ (  ab ( x 4 = ( boundary a b + u ˆ − d P ) ) ∇ u a 2 ( ` ) 2 ` U using ( d ` u ) ) on it scales inversely with ∗ ( d 1 2 ): 2 u x ) is simply ) of twistors modulo terms in u x 2 d boundary points ( ` ( ` ) d 2.8 ( ( x is a special case of ( P ∗ ∗ ( 2.14 ) = ). both lie on the same connected spacetime, and have the same Weyl ` u x P P ` = ( 2 two ( µ b d a ), but with opposite handedness roles [ x SO(3). Twistor space now acquires only U ab x I P = − P  covariant derivative ( U 4 4 ), and vice versa. The corresponding measures are related via: P and ` d ( µ ∗ x P EAdS ), which is consistent with their corresponding metrics/measures. In intrinsic ` ( 4) down to , P Finally, if we fix Now, consider a choice of The quotient spaces: elementsclasses of in Multiplication by and boundary terms, this operationthe spinor is metric just index-lowering from spinors into cospinors, using defined analogously to ( particular, the measure ( by the quotient space The spinor metric on into the second position. SO(1 — the space spannedmetric. by From the the point matrix of view of the 3d boundary, the appropriate spinor spacecovariant derivative using with spinor indices as: so that the left-handed spinor index is forced into the first position, and right-handed one left-handed spinors at being simply indices: it is againinto the the flat tangent derivative space at spinor spaces case of a bulk point can be defined equivalently as: The decomposition signature, JHEP10(2020)127 , the (2.21) (2.25) (2.22) (2.23) (2.24) (2.20) 1 2 − ). More generate b Y = ( ); Y . Since the l f l a ( µ ) implements · . Y γ P Y ` ( ∈ δ iUY as instead of an anti- in Clifford algebra. . − ] µ e l`U ν ab ) , l γ ). Another important iI = µ µ U iUV [ Y ` ( ) − ( γ l ( e = 2 u , g ) Uf ) ? 4 ] V ) simplify into: b d . Y . ( + ) ,Y f ) Z ` ]. a b ( . 2.19 Y . Higher-spin algebra is just an 2 Y Y instead of a vector ( du g − µν ab ) = a ) ( ) η ` π , a ( iI Y f Y 2 2 U ( ). A star product with − + du ). The quadratic elements (0) + b ? f f ) = ab 2.5 = l ); ) Y Y Y ` ( 2.22 a ( ( Y } = ) = ( Y ν P ) δ V f ? δ l Y – 11 – ∗ ( 4 , γ ( = ) ∈ u µ f b ; 2 Y γ ? Ud ( d { 4 4), in analogy to the role of ) for even functions `lU tr , d ?Y ? f iUY a = 4 = e ) = 1, and commutes with even functions ) Z ) ) Y ) ` ` Y g ? f Y ( ( ( ), as defined in ( ( ( U u u ? ( δ ) = Y 2 ( ? δ d Y δ ) and measure relationships ( ) Uf ( 4 ; Y d ( ? g ) = tr ) l δ 2.13 ) ( ) respects a trace operation, defined simply by: Z u Y ; 2 )–( ( ) imply: d f ? g ) f 2.22 ) l ) = ( ( ` ? ( Y u 2.12 2.24 u ( 2 + ) with the associative product ( d ) ? δ Y 1 4 ` ( ) ( − u f Y ( = = f U U The product ( 4 ) squares to unity d Y ( δ generally, eqs. ( which satisfies tr object is the delta function a Fourier transform in twistor space: the spacetime symmetry group SO(1 The HS symmetry algebra isfunctions the infinite-dimensional Lie algebra of even (i.e. integer-spin) commutative product: This can be extended to arbitrary functions (and distributions) via an integral formula: 2.2.1 HS algebra Consider again theapplication Clifford of algebra the sametwistor metric concept, is but antisymmetric, to wecommutator. get a Imposing a twistor a commutator [ symmetric ordering convention, we arrive at the following non- 2.2 Higher-spin algebra andIn the this Penrose subsection, transform we introduceform, some in a fundamentals of condensed HS version of algebra sections and 4, the 5 Penrose and trans- 8B of [ decomposition ( and vice versa. If we fix the relative scaling of the null vectors JHEP10(2020)127 (2.33) (2.32) (2.28) (2.31) (2.29) -point n , pick out ): 4 , 1 . R 2.30 , , ! ∈ ) ), which will be 0 ξ µ Y Y n ). ξ ( · ` n 00 ` · 2.31 ξ ` 1 1 ; (2.27) ` 2.10 2 − ; (2.26) ? δ 2 Y ) iY ` 0 ) ) Y/ Y ξ ξ 0 ( ) Y ξ · ( ( iu 0 ` exp iu − ξ δ e e ) . iY ξξ ) )( ) ) 00 ; (2.30) ξ ) ) ) − ` +1 ξ ξ · ( ( p · Y 0 ` ξ Y ( u u 0 ` 0 . ( 00 ξ · ) in this case is over the spinor space ` 2 · ` − + + p − · )( ` p δ ab ` 0 n 2 ) ` ` 2 ) Y Y iY ``

2.11 ` i · 0 ( ( 0 ) is by first expressing it as a single spinor − ` ` ` f f (  ) = · ) ) exp ξ ξ 2( − Y exp ` 4( =1 ( ( ( 0 2 0 2.31 n p 0 ξ u u − ` ξ 2 2 `` 2 · Q · ( d d – 12 – s ? δ i ` i ) ) ξ q ) ξ ξ for manipulating spinor spaces, one can work out −  ( ( = − Y P P ( ) Z Z 0 δ ab ) = ξ ) = ) = A taken directly from the 2-point product ( · Y 2.1.3 2 Y Y ( 0 ) = ) = ( ( ) = n ξ 0 A 00 ` ` ). Here, the 3-point product reduces to the 2-point product: Y Y ` Y )( ( ( ( ξ ξ ? δ ? δ · ) ,... ? f ? δ ) is thus the result of an imaginary Gaussian integral over a ) ξ ? δ µ ) ) ( 0 Y ) Y ( ). The Gaussian integral ( ( ? . . . ? δ Y Y ` , ` 0 p Y ). The resulting integral this time is a Gaussian one, which can be ), and then evaluate that integral using eq. ( 2.31 ( ( δ ` ) µ ( ξ ξ ` f δ δ Y 2.11 ? δ ( ) recursively, we arrive at a closed-form expression for the ) = 2.26 2 below. 2.26 ) ` Y 4 Y ( ( 0 2.31 ? δ ` ξ ) associated with these spinor spaces have the following properties under the ) δ Y Y ? δ ( ( ) are null ( ξ ) 1 δ ` Y δ ( ξ ,... δ µ 0 ), with the quadratic form Applying ( Star products with any additional delta functions will continue to result in Gaussians. 00 , ξ ` µ ( ξ product: The 3-point productcomplex ( spinor space.crucial This in is section the reason for the sign ambiguity in ( where we note that theAgain, square the root best way as to written evaluateintegral is the using product real, eq. ( i.e. ( theevaluated quantity using inside eq. it ( is positive. P The most efficient way tointegral derive this using result eq. is ( to first express the product asA a particularly single simple spinor case( is when all points are on the boundary, i.e. when the vectors Using the techniques fromthe section star product of a pair of spinor delta functions: Recall that choices of spacetimesome points, subspaces i.e. or of quotient embedding-space spaces vectors functions of twistor space, which describeHS local star spinors. product: The delta 2.2.2 Star products with spinor delta functions JHEP10(2020)127 . ) . ), ), x ’th x s ! − ; 2.35 2 ( 2.29 (2.34) (2.36) (2.37) (2.35) ). The P ! Y/ 0 Y 2 ∈ 0 ; . follows the ) ξ x 1 0 Y/ x · ( i 0 ` ξ − iY ξξ ξ · C ( ≡ y ξ − − . ), which encodes iY ξξ ) − +1 0 ) Y Y x n ξ ) ) ; − ( 0 0 totally symmetrized ` into its Weyl-spinor · y x ξ ξ ) ). Similarly, the left- 0 ( s Y ) x · ξ x ( − ) x 0 0 ( C ( iu ξ )( − ξ ( P e ( P ξ )( y ) · ) P ξ ξ ) ) ξ · x + ( ξ − ) with 2 − ) ξ ( ( ( ( x x p y ( ( s y , p 2 + ) iMP iMP ) = Y ...a ) . x master field 2 2 ( 1 s ( ( ) to solutions of the free massless field a x Y u

) and a left-handed one ( C Y x ( ? δ ( F ) ) = 1 ) F exp exp untouched: P ) where one of the delta functions is either x ( Y Y 0 0 ) ( ( ∈ ξ ξ x u x · · 2 ) − 2.29 ( x d iF ξ ξ totally symmetrized indices in the left-handed ( -dependent – 13 – y ) ? δ y x x ) s − − ( ) is easy to invert, since, as a special case of ( ) = ) ) P is an object Y bulk. We again restrict to integer spins, i.e. to even 0 0 ). The field strengths are now encoded as the 2 ( Z Y ξ ξ x 4 x x i ; · · δ 2 2 2.35 0 0 x , as well as their derivatives. The factor of − ) has no spacetime dependence whatsoever. Through the ( ξ ξ ( x ). In HS language, the Penrose transform is just a star ) has 2 ). To extract these field strengths from the master field ) = C )( )( P x ) Y x ξ ξ ( ( x ( EAdS 2.35 · · s ∈ − 2 F ( while leaving ξ ξ ) (0) ) y ( ( x ) and shifting the integration variable, the star product in ( s ) ...a C x − ): 1 − ) in ( ) is the delta function on the right-handed Weyl spinor space at ( + p p ( ( a ) = y y ) Y Y C 2.26 x ( ( Y ( 2.26 x y F ) = ) = − δ field strength at ; ( ). For spin 0, there is no right-handed/left-handed distinction, and we Y Y x s ( ( F x ( ) and 0 0 ]. The transform ( ξ ξ x − C ), it is converted into an 2 ( ( into a right-handed spinor ? δ ? δ P P ) Y
2.35 ∈ M ) x iMY ( − e ), we first decompose the twistor argument y ) Y Y Y ( Also useful are the generalizations of ( ; ( ξ x ξ δ ( δ handed field strength spinor space have just the scalarC field pieces Let’s now describe howright-handed spin- the various bulkspinor master indices, fields all are of encoded which inside lie in the right-handed spinor space The twistor function transform ( the higher-spin field strengths at conventions of [ we have: and Fourier-transforms where we recall that More explicitly, using ( decomposes The Penrose transform relates twistorequations functions (of all spins) intwistor the functions product of the form ( 2.2.3 The Penrose transform shifted, or multiplied by an exponent: where the product inside the square root is understood cyclically, i.e. JHEP10(2020)127 (2.41) (2.42) (2.38) (2.40) (2.39) . a Y . ) ); , x x ( (  ) are identified ) ) ) k k x + + ( s s s 2.38 2 2 b s ...b . ...a ) ...a ) ) +1 s +1 ) 1 ; x s -point correlators of the k s k b s − b ), we conclude that the 2 ( 1 − a ( − n massless scalar. s ) a ( ( a 2 x ) can be combined into a C C a ( ) C k k a 2.24 a ) ); . . . y k k 2.38 x ) b b . . . y ) + , ): ( x ) = 0 ; ) x ˆ ˆ ) . 1 1 ∇ ∇ x x − ( a a ( ( (0) ( s Y ] b y y ( 2.38 s C ...... ) ) (1) ) as a Fourier transform in νρ 2 1 1 x x conformally ? δ ...a a a C ) describes a Maxwell field strength for Y ( ( ) = 0 − ( ( 1 ; ) µ s s b x 1 1 [ ) 2 2 b b x 1 ( s Y ( ( ) ( ( a s ∇ s ; 2.39 ν ...a ...a ) = ) ˆ ˆ ). A particularly simple case is the master ∇ ∇ C ) which don’t appear in ( C s 1 1 x s − x ( ( a a ( µ  k k ( Y i i ) = ··· s C C s ; C 1 b ν x ). Recalling eq. ( 2.35 (0) x ν s s – 14 – = = ) ( )! )! ( 1 a µ s C s s ( 1 1 µ C ) = µ (1) µν =0 =0 ) is that of a 2.28 C (2 (2 Y = 2, and their generalizations for higher spins. The ∇ C 1 Y Y x . . . γ ;

( µ µ µ s ) =0 =0 x 1 k 1 ∞ ∞ ) x s s ∇ ∇ ∇ b ν X X + (0) x 1 1 − s − k ( a − 2 µ ( b C ( , which takes the form: b γ C µ ) = ) = s 2 : ) ) x 1 x x 4 ( ) − = 0 : = 1 : ≥ − y . . . ∂y ) ( . . . ∂y ; ) s s s y ) : x x ; ) = ) automatically ensures that these fields satisfy the free massless x acts on a master field 4 , respectively: ( x, Y − 1 ) x x − 1 x, Y ( ( b ( ( x ( x b C ( point s − C C ν ; 0) is the spin-0 field. Eq. ( 2.35 ( ∂y ) C s ∂y ) k y k x ) µ k k ( EAdS + x + → − a s ··· ( ) s + 2 C 1 s x a x ( ν 2( ) and 1 2( s ∂ ( µ ∂ ) ) = x C antipodal . . . ∂y ( x ) y ( . . . ∂y 1 x a ( ) 1 (0) x a ( ∂y C ∂y Finally, those Taylor coefficients of = 1, a linearized Weyl tensor for antipodal map 2.3 Boundary correlators In this section, we constructfree the boundary HS-algebraic CFT. expression for the field equations or thefield Penrose at transform the ( thanks to the star-product identity ( Knowing the master field at one point, we can calculate it at any other, using either the via the Penrose transform as derivatives of the fields ( Note that the field equation for while s Penrose transform ( field equations in The right-handed andfield-strength left-handed tensor gauge for field each spin, strengths via: ( powers of JHEP10(2020)127 ) ) ) µ ], s = ` ` ( ( 1 s 1.1 A 33 , − ). We (2.45) (2.47) (2.44) (2.46) (2.43) ...µ )  1 32 s ( µ and 1.1 = J . ) from ( s ( G s ), or, in the ), coupled to J   0 . is N ...k ) 1 s ) ( I k (2 `, ` ` . ( ( J φ O I ) ), we are assuming ` φ ( ) s ≡ O , ` 2.43 0 ) ...µ ` ` ) 1 for the sake of uniformity s ( µ ( I J )Π( φ 0 N ) ) : ]. In ( ` , 0 2 ` ( ) + permutations µ ` ( ) I ( 42 s ` λ ). ¯ I ( ... φ +1 . n ¯ s p φ ` ...µ have Bose statistics, but switch to ) 0 3 1 s , ` ` ...µ I ( µ d ) p 0 · as 1 1 s ¯ bulk [ ` φ ` ( A µ ` ( 3 I 4 J 2 , . . . , ` d G s 1 1 is the conformal Laplacian. The internal =0 ∞ , keeping in mind the orthogonality to − µ ` s X s dS and Z n =1 `  √ 0 are morally similar, with the addition of Y I , the correlators read: p 3 π I − φ d 4 . . . λ   φ . . . µ I . We can upgrade the indices on 1 I n 1 − φ Z s s > µ ¯ – 15 – φ , and “+ permutations” denotes a sum over the , and µ 1) λ  1 1 2 − ...k ` ) I linearly independent; in this case, the sum is over = 1 − s I ¯ ) = φ ( ( k 0 I φ ≡ with ` ) = ¯ A (0) φ N 3  ) `, ` J d s I ( +1 ( are not independent but linearly related, then only the indices = `, λ ¯ φ n . The fields ( G J Z 4 ` I ) ` and , s ¯ 3 1 φ ( − I d -proportional shifts. The action of the vector model can now R J φ µ ...N ` Z and )] = connected − into I E , ` ) 0 s φ = ` n ` ( = 0. [Π( . . . k . If CFT -point correlators of the free vector model consist of 1-loop Feynman (0) 1 ` s have conformal weight ), where we redefined the range of S n k · are replaced by bilocal scalar operators I CFT N ¯ ): λ φ ) S s ...J , ( , ` I ) = 0 J 1 ` φ ` λ ( ). More covariantly, we can write: · takes values from 1 ) vector model, with λ (0) I πr 1)! cyclically inequivalent permutations of ( J N case, Sp(2 The correlators of currents In a flat conformal frame, the propagator of the fundamental field For notational efficiency, we will package the (symmetric, traceless) currents (4 D / − 4 1 n can encapsulate them all atwhere once the by switching temporarilysources to Π( a bilocal formalism [ Here, the product is( cyclic, i.e. some derivatives acting on the propagators, after the pattern of the derivatives in ( The connected diagrams. For the spin-0 operator where − of the results below. into their scalar contractions with a null boundary vector Fermi statistics if we analyticallya continue U( to a all integer spins even-spin currents are non-vanishing.dS The internal symmetry is then The fields index Consider the free vector modelcoupled linearly on to the external 3dfrom sources boundary, 3d with indices the currents and the equivalence under be written as: 2.3.1 The CFT picture JHEP10(2020)127 . ) is ) by ` (2.48) (2.49) (2.50) = 0 ` 2.49

2.48 ) 0 ,   `, ` ( O ) s µ 0 ∂ ) corresponding to lightcone, while the x ∂` ( 4 , 1 ) conspire to make the (0) ... R C Penrose transform of the flat derivatives in ( 2.49 +1 , m 4 ∂ ) + permutations , µ 0 1 x +1 off the R · ∂` p 0 ` m )). , ` 2 µ iY `xY 0 p ∂ `, ` ` ) to ( ∂` ) — the unique twistor function that 2.35 G 1.1 exp Y ( ... x ` n =1 δ Y 1 · p 2 µ ` (   ] starting points. In our present framework ∂ − n . ∂` never take µ 44 – 16 – 1)  , µ α` − ) = s ∂ ). Here, we will complete the picture by presenting m ( 2 µ 2 31 Y + , λ ( N  x µ 2.47 30 m λ = ? δ 1) ). Thus, the spin-0 bulk field ) ], though the authors there stopped at bulk master fields, → − ( Y µ which correspond to the boundary sources (here, we are ba- 31 ( λ ` 2.29 ), which we can express covariantly as a differential operator =0 s δ ) and the combinatorial factors in ( X 0 m  connected ) boundary sources to the spacetime-independent twistor language.

s 0 ) 2.44 µ `, ` ] and bulk [ 0 n ( `, ` ): , ` ( O ), ( 0 local n 43 . It must be a multiple of . . . λ , O ` ` ). The correlators for the local currents can be extracted from ( ), which is indeed (up to an imaginary prefactor) the boundary-to-bulk  1 ( 1.1 `, ` sources, as in eq. ( . To apply the Penrose transform, we will need the star product formula: ) x µ ( 34 s ` O · ( λ O s ` D i ( ... , . . . , n ) ], which we will then use to construct the dictionary for nonzero spins. i/ ≡ 0 1 2 bilocal 2 ) = , ` − 1 ` `, λ ( ] for ( ) acting on 2 ) is It’s easy to guess the twistor function that should correspond to a spin-0 operator at Traditionally, twistor functions are more closely associated with bulk fields (via the O s ( )

Y s J ( ( ` depends just on which is a specialδ case of ( boundary-to-bulk propagators sically following the logicnot in taking [ the final step of the Penrose transforma ( boundary point We will begin with spin 0.results of The lessons [ of the spin-0 case will enable usPenrose to clarify transform) the bilocal thanfollowing with way: boundary the quantities. twistor functions Thus, we’re it’s looking helpful for to are think the in the then to combineboth them boundary into [ HSof invariants. embedding space In andout the spacetime-independent [ twistors, literature, this this procedurethe has has dictionary from been been carried done from result invariant under shifts 2.3.2 The twistorLet picture: us spin now 0 translatefirst the to CFT replace correlators the into CFT HS-algebraic sources language. with The HS approach algebra is elements, i.e. twistor functions, and Note that the switchlegitimate. from In 3d particular, flat derivatives conformal derivatives in weight of ( plugging in eqs. ( D where the product is againtations cyclic, of and (1 the sum is again over cyclically inequivalent permu- The correlators now read: JHEP10(2020)127 ): , ) ), we (2.57) (2.56) (2.54) (2.55) (2.51) (2.52) 2.54 +1 ) in the ], let us p propaga- approach 2.46 +1 , `
p n ) -dependent p 31 ` , n , ` Y ( p ; 30 , ` G n 4 as in ( ( ` bilocal ) into ( . 0 bilocal insertions − ( Y n G =1 ` 0 . Y ) p · n (0) ) n ` 2.54 +1 ) 2 p +1 π iY `` p 4 , ` . 0 p , ` − ; (2.53) ` p ) with (  ), but with the ( exp ` 2 ) ? . . . ? κ 0 ( G − x ) ` n · G · 2.48 ). To remove the 2.46 ) Y n =1 1 ` i ` ; Y p ( n =1 1 2 1  Y p 4 π ` . ). Plugging ( − ( 2.46 2 ). Now, following [ 4 − ) x √  − (0) ( Y = 4( = π κ ( 2.31 ` =
) (0) δ ? ) =
C )) = ) π +1 + permutations tr i Y ) p 4 ; Y Y `  Y 0 n ( ), up to the same factor of ; ·  ; 2 x 0 n +1 , ` p ` − ` n ` ). Crucially, this matches (up to normalization) ( n ) as drawing the propagators ( n ? δ – 17 – -point invariant then read: 2 ` 2.48 ) ) = 1) ) ( i ) n (0) 0 (0) − Y − (  Y 2.45 2.54 ; ( ; ` `, ` ? κ ?K 4( ` =1 ( ( 4 ( ) N n p G (0) Y ··· Q (0) = ; κ , via the identity ( ? . . . ? κ ? ` ) and ( iκ q ( ) ) ( local spin-0 insertions, as in ( ), we define the twistor function corresponding to a spin-0 ? Y Y (0) = n ; 2.33 ; tr κ 1 0 1
` ) 2.51 ( , ` connected Y 1 ), the HS invariants constructed from these functions immediately ( E ) = ` (0) ) n ( κ Y ` n ; ` K 0 ( 2.54 ? ), we can quickly recover one of the main results of the one at a time ? tr (0) `, ` ( tr 2.54 K ? . . . ? δ as: ) ...J ) removed. Thus, if we define the twistor function corresponding to a bilocal ` ) 0 p Y 1 ( ` , ` 1 ( p ` ` δ ( (0) ? G J ]. The key observation is that the Feynman diagrams ( 2 D tr then, by virtue ofevaluate ( to the Feynman diagrams from ( tors source as: 2.3.3 From spin 0Starting from to ( bilocals toin all [ spins are just like diagrams with 2 obtain the spin-0 correlators in HS-algebraic form: We can think of theFeynman star diagram products in ( The corresponding scalar bulk field and which follows directly from ( with the Feynman diagrams encodednormalization in factors the in correlator ( ( insertion at combine a sequence of these twistor functions into the unique HS-invariant trace: propagator for the conformally massless scalar JHEP10(2020)127 , ) ) ) )] Y ; Y x 0 ] 2.63 2.62
· (2.64) (2.62) (2.60) (2.63) (2.59) (2.61) ) 0 `x ` `, ` ) Y ( ; )( x , 0 n · x K 0 ·  , ` ` ` ), and obtain n ` . ( ), is given by: + 2( ) 2.62 + 2( 0 x 0 `, λ ` · `` ( [ · ) ` s . ` ) ; (2.58) ( iY iY `xY 2( ?...?K 2[ J  Y ; ) 0 2 Y exp + permutations exp ; , ` )
0 1 1 , ) x ` )] ( , ` · . ) that are null-separated, which  x Y 0 1 0 ) · ; Y ) ` K ` 0 0 0 to the resulting bulk fields, and n ( Y ) ` `, ` )( ; 0 1 ) 0 K , λ x `, ` )( s ( ( , ` n · x iY `` ? ` 2 `, ` · ` G D ( ` ( + permutations tr ) ( 4 ` 4( n 1  K s G − ) is very singular. Instead, we will first (  ∓ ) +1 Y s + 2( ( n exp ; 0 0 ) = ) = ). It remains to evaluate eq. ( ) ` D 1) Y Y · x – 18 – ), apply `, ` ; ; − · ` 0 ( ( 0 2 2.49 0 Y ? . . . ? κ ) = 2[ ` ; K 4 = 0. Specializing to this case, the master field ( , ` ) 0 N `, ` ), describing an insertion of Y 2 )( 0 ( p Y ; ` 1 ` Y x ( ; · = π `, ` ; K · ∓ 1 ` ( connected ? ` `, λ ( E ( K , λ ?K tr `, λ ). Therefore, the correlators for all spins can be expressed ) ) ) = 1 ( s ) n ` p ) ( ). It is clear that the answer should be the Penrose transform limit of Y ( s π κ Y ) ( ( 0 Y , λ 1 2 ; 2.49 x ) of bilocal operators take the same form as the spin-0 local ` ) only probe pairs of points ( κ ; connected s n 0 1 (

` ) ( κ ? δ , ` = ) 0 n `, λ 1 ) = ) 2.48 n 2.49 ( ? ` s ` ) , ` ( Y ( Y s tr n ; ( ; 0 ` K x  κ ], as the master field: ( ( 2 0 ...J O `, ` +1 ): ( n ) `,` 1 C ... 1) iK ) , λ 2.55 − boundary-to-bulk propagator. Unfortunately, it’s difficult to evaluate ( 0 1 1 ( ≡ ` the differential operator from ( s , ` ( 4 ) ) N 1 ) 1 ` s Y s ( ( ( ; = x J D O (

D 0 `,` that the derivatives in ( translates in embedding space into becomes: C where we again have a sign ambiguity due to a Gaussian spinor integral. Now, let us notice directly, because the take the Penrosefinally transform Penrose-transform of back intowas twistor evaluated in space. [ The Penrose transform of with the explicit form of of a spin- where the twistor function Now, recall that correlatorsbilocal for correlators, the via local eq.in ( currents HS-algebraic of form all as: spins can be extracted from the correlators ( Thus, the correlators ( Here again, each individual propagator can be attributed to a star product, according to: JHEP10(2020)127 ) of x · j : 0 ` ` (2.67) (2.66) (2.70) (2.69) (2.65) exactly , which ), which 2 λ ), we can ) or ( 2.65 x 2.49 · ; (2.68) ` , ) j x . ; ) on a purely right- · ,  1 ) ` ) ) ) j ( x ) x ) x x − . π x − x 2.64 · s · · ( ( ( 2 ) 0 0 0 y y s y ` ` 0 ` 0 x 0 )  ) · )( ; )( ) will turn it into )( `` `` `` x will act on ( +1 ) ) ) µ x x λ x = j must be acted on by − s x +1 x x ( · · · ( λ ( j ( ) s +  j 0 − ) 2 s ` ` ) ` ) ( iy x ) ) . iy ) +1 ) − `` ( ): s x x 4( x 4( x 4( iy can contribute, and in those, `λ y 2 ( x s · − ) · ab ) (  x s ) `  ` x y `λ y ! ( ( ( − · ) 1 ; exp exp j ( `λ y `λ ≤ x ), i.e. of a boundary cospinor at span a totally null bivector, which can ) P x ` y ( ). We expect that only the terms with ) ) ` x ( ( ( j ) y ( 0 ( µ x x =0 ∞ ∈ x ( s = ( y j ! X · · λ P `,` − ( s x 0 0 . Finally taking into account all numerical ! ν ( ) 1) ` ` s C  πs y ∈ λ x 1)  − y 4 ; s,j πs )( )( µ · ( a and c = − 1 1 ` x 4 0 x x , and this is easy to check explicitly. First, note ( (0) ( = `  m · ·  0 µ j ∓ ∓ – 19 – ab µν s ) =0 ` D  ` ` )( ) j γ X `,` Y vanish. Thus, they can only survive if first acted ( ( = 1 x x 0 C = =  ·  ∓ = = ) is arranged such that the result is invariant under p p ( )   ` `` ) ) can satisfy this only if, as expected, the only term b y  π π ) ) ( x ) ; ) ) disappeared: it contributes only to the field strengths’ ) 2 2 (and similarly for the left-handed field strengths). Now, ) m x x − x p ( x 2.49 ( ( a ( ( factors, and the other y 0 y y 2.67 π s,j y ; ; ; m 0 2.64 2 ) = ) = c ; `,` x ) ) x x ) are contained in the evaluation of ( . Moreover, each factor of x `` ( ( ( x x C ) 0 ( 0 0 ( s 0  − y ( ) ( ) = `,` `,` `,` s ; ) `,` y . It is convenient to Taylor-expand the exponent in ( 2.38 ( D x C x C C ) ; ( C s (    D x 0 (  y ) ) ( ) s s ; 0 s ( ( (0) `,` D ) as: ( x ) ( `,` C D D D 0 x D ( C y `,` ; ), not to the field strengths themselves. We are now ready to apply the C . The r.h.s. of ( x µ ( , the matrix products 0 0 0 : 2.41 ` α` `,` = 0 : C = + ) s s > s ` µ ( λ derivatives will act on D will contribute to derivative, since two derivatives (when contracted with s s → = µ We can simplify further bybe noticing re-expressed that as the square of a twistor for some numerical coefficients recall that the combinatoricsλ in ( with a nonzero coefficient isfactors, the the one result reads: with factors. Therefore, withoutwrite keeping track of the combinatorial details in ( that at on by the derivativesone in again vanishes. This meansthe that only terms with and similarly for thej left-handed part Note that the last exponent inderivatives ( ( differential operators directly corresponds to an expansion in spins: handed or purely left-handed argument The bulk field strengths ( JHEP10(2020)127 , ) b a ` 2.74 (2.74) (2.76) (2.77) (2.71) (2.73) (2.78) (2.72) , ) . . , Y ) ( ) x ; (2.75) ` · δ x x · · ` x  . ` ` · s iY `xY 2( 2 2 ], and in our present s iY `xY ` 2( ; iY `xY a 2 2 . In intrinsic boundary ) +1 44 iY `xY s ) ` s exp 2 2 x +1 ) exp s ) s exp s − ) 2 2 2 ( x s x ) exp )  ∂ ( . . . ∂Y 2 · x ) 1 , with an extra factor of 2. In Y  ` Y · my a a ( x Y ) ( my ` Y x ( ! − x ( m ) − . ! ∂Y x x − 1 πs x s ( ( · boundary-to-bulk propagator. To- ) 1 4 · b +1 πs +1 ` s 1) mP s M`P s 4 ` 2  2 ) − ) .  `M x is a square root of the null polarization + ( x = + ( ( b ), we conclude that the master field ( iM`P iM`P · · a ), i.e. a spinor at s = s a + (  ) 2 ` ` 2 ` )   M  s ) ( ( ) ) ): ( b 2 ) M ∗ x 2.74 ) a a ), we must rewrite the polarization twistor Y Y `M x − ` P x x ( ( exp exp 2.73 y y – 20 – = x x ; ; = ( 2.72 ...Y mP · · 2 2 a x x ( ν 1 ), ( ( ( ` ` M`P 0 0 a ! m λ ( Y µ − − ! `,` `,` 1 ` πs  ) becomes: 2.70 C C 4 1 πs ):   ab ]. s µν ) ) 4 2 ) = ) =  γ s s a ( ( 39 2.72  Y Y = ( ( 2.34 D D ! x x  = )  πs ...M ? δ ? δ ) Y 8 1 ; )
) then become: Y and comparing with ( a ) x ; a ( 0 : M 0 x Y ( ( iM 0 2.69 ` M + `,` δ  s > `,` C Y  ( C ) `  s iMY ( ) δ ) = s e = 0 case is now included. Note the reappearance of the last exponent ( D Y , the master field ( ; D a s is just the raised-index version of the cospinor ), to encode the fields’ derivatives. This master field for boundary-to-bulk prop- is an arbitrary twistor; however, since it enters only through its product with ) as: a , in the sense defined by ( M `, λ ` a µ ( ( ) M λ s 2.64 M P ( κ To find the Penrose transform of ( ∈ a where we recall thatvector the polarization twistor Taylor-expanding in is the Penrose transform of the following twistor function: We can now find theof inverse Penrose the transform star-product of formulas this ( master field, by using a special case terms, terms of Here, it is really an element of the quotient space where the from ( agators was first writtenembedding-space down formalism in in an [ intrinsic bulk formalism inm [ As expected, these aregether the with their field derivatives, strengths they of combine into a the spin- master field: The field strengths ( JHEP10(2020)127 ) ) ], ), ), 2 4). Y , ; ) for 2.55 2.62 2.31 ) from `, λ Y 2.77 ( ; ) s ( polynomial ) are similar κ `, λ ) that allows ( ) s Y ( 2.61 ; = 0 case. κ ` ( ) fails to define a s in ( ) recovers the spin-0 ) propagates to the ). Finally, the same (0) ) ). However, that sign s κ . 2.22 ( ) becomes a boundary Y κ 2.77 4 2.31 ; 2.52 ): in the limit where the 0 2.63 ) for infinitely many `, ` in ( 2.31 ( K (0) = 0 in ( 2.52 ), this sign choice gets squared, κ s will also appear in star products of unambiguously defined for polyno- 2.56 possible. In other words, we can have is (0) κ is -point current correlators in terms of HS n ). In ( sign is meant to be equal to that in ( correlators, thanks to our construction ( s 2.55  – 21 – ). ) ends up translating into the sign choice ( ). In the course of our derivation of ], where we considered only bilocal correlators, we ) of all the Y 2 Y ; 0 ; 0 2.63 2.61 ), and the signs as we wrote them are then consistent with `, ` ( `, ` ), the particular Gaussians that arise from their star prod- ( K Y 2.31 K ; 0 `, ` ( ), but with additional polynomial factors. These star products are K ) from which we started. The star products of 2.54 ), and from there to the sign choice ( ) which defined the Penrose transform of 2.52 ), then a consistent choice of signs in ( 2.51 2.63 ), and then stick to that choice throughout. This is what we have done sign is meant as its opposite. The sign choice in ( approaches a boundary point, the star product ( (0) 2.61 x κ ) from the bilocal ∓ 2.31 ) via a bulk detour, we encountered a seemingly separate sign ambiguity, in the )–( Y ; Y ` ; ( 0 ) 2.60 s [NOTE: in our previous work [ To have such a consistent algebra, we need to make a choice of sign in the 3-point On the other hand, the star product certainly ( , which is in turn consistent with the, one and we it’s had consistent for to resolve them by following the pattern of the `, ` ) ) κ ( s s -point trace ( ( ( κ sign ambiguities that appeared inκ star products of made a sign choice that is inconsistent with the ones described here. It involved a different K star product ( choice is actually of abulk piece point with the one we3-point product already of made the in form ( one ( another. The sign choice in ( implicitly throughout this section:and every every n us to express theproducing spin-0 a correlators definite sign as for in ( the bilocal insertions ucts, and finite superpositionsfrom thereof, coincident as (or, long more as generally, null-separated) we boundary exclude points. singular productsproduct that ( arise mials, which span theinconsistencies space can of only all arisebasis functions. functions. when Similarly, taking as From superpositions longand this as ( of our point only of interesta is view, consistent in ambiguities star-product the algebra and boundary for correlators the ( functional class of the local insertions can’t be resolved consistently with theIn topology other of words, the once spacetime symmetry weone group allow that SO(1 a sufficiently includes broad allconsistent class algebra. possible of This twistor Gaussians), obstruction functions then will (in play particular, the a star key role product in ( section 2.3.4 Fixing theLet’s sign now factors address thesection. sign As ambiguities we thatspinor pointed spaces, we’ve out, which been come these up arise carrying inthese the sign from along evaluation ambiguities imaginary of can throughout never certain Gaussian be star this resolved integrals products. without over some As inconsistency. complex discussed In in particular, [ they algebra. As a consistency check, wetwistor note function that ( substituting to those of guaranteed to correctly reproduce theof spin- This completes the formulation ( JHEP10(2020)127 (3.5) (3.2) (3.3) (3.4) (3.1) ) more 3.1 ) via: . ` , , (  s N N n s ! ...µ ! ) that represented ] 1 ] s ) ( µ ...µ ) n G 1 s s ( ) A ) ( µ σG κ G ). We then have an connected A σG E ) =0 tr(Π tr( . n ∞ 2.49 n e e X ` s ) . ( n ). In particular, for a spin-0 n )–( det[1 + det[1 + Π Y ) ` s ` ; 3 ( Y

`

I ) d ( ; ...µ n s ` 1 2.47 φ = = s ( µ ( Z s ...µ   ) 1 s

...µ µ ( ... ) 1 s ...J G κ sources can be obtained as a Taylor 1 µ ( s ) ) σG Π ), which amounts to choosing the sign κ 1 ` ) ( ` s Y ...µ − − 1 ( s µ 1 ; ] s ] 1 ( µ 0 2 local s ...µ G ) A 1 , ` s σG ) ...µ 2 ( µ . . . λ 1 1 ` =0 s 1 A ( µ ∞ ( – 22 – 1 µ X s J ). To encode compactly the sources for all spins, λ 1 =0 ) with ∞ D s boundary limit combined with a limit of coincident ` X ?K ) are tempting us to integrate them into an HS- ln[1 + 3 ≡ ln[1 + Π × ` ) 3.1 d → ) 2.45 3 ): Y tr d Y tr 2.61 ; Z ; 0 1 3.2 1 ! N Z 1 1 N n , ` ), ( − 1 `, λ − ( `  ) =2 ( ) = ∞  s X n ( 2.55 Y K ) would be constructed from the correlators as: . For the free vector model, we can write out eq. ( κ ) are included in the algebra. The reasoning that was given for (  ` i ( ) . Y F s ` ` ( ( ` δ ...µ = ) )] = exp ] involved a bulk (0) 1 s )] = exp 0 ` ( µ 2 ` J ∼ ( h , ` A is the propagator ( 0 σ )] = exp ) [ ` ` ] to agree with the signs in the present paper.] 1 ( Y 2 ) ; − s ` [Π( ( local  ( Z A [ (0) = κ ) differently every second time. This becomes inconsistent once local insertions bilocal G insertions: local Z Z s On the other hand, we can perform the analogous construction within HS algebra. We 2.31 We can now assign a twistor function to any finite source distribution from which the partitionexpansion around function ( begin by unpacking the polarization indicesspin- from the twistor functions it’s best to embedexpression formally them identical again to in ( the bilocal formalism ( source we have: where Here, we ignore for1-point simplicity function the 0-pointexplicitly: function, since and the renormalize patha integral away functional is the determinant Gaussian, over divergent the the space correlators of will boundary arrange fields themselves into tional of sources 3 Conflict: Euclidean partition3.1 functions From correlatorsThe to partition correlator function formulasalgebraic ( partition function. In standard CFT language, the partition function as a func- in ( such as the sign choice in [ points. It appearsnow that updated the [ inconsistency arose from the ordering of those limits. We’ve sign for the star product JHEP10(2020)127 ), Y ( (3.6) ` iδ ) of the  Y ( F , ), and instead ). At the very ` 4 ( Y s in the exponent, N/   (  1 4 ). In contrast, the F ...µ ) )]) ) 1 } µ s − ( µ λ Y Y ( ( A . Now, let’s get to work. F ) comes out positive. Also, ) is formally very similar to 1.3 [1 + {z factors ? 3.6 ?...?F 2.53 n ) Y ( ), we can simply extract them (or, ` = (det F | ( s ) over source points and summing the   ). For simplicity of the final result, we ? )] ...µ ) 1 tr s Y 2.61 ( µ ( ... ( A F ? +1 actually look like, when we expand the twistor n ), apart from a factor of ln n ? – 23 – 1) ). HS [1 + 3.3 − ? Z ( ln )–( 2.54 modes ? =1 ∞ ). Let’s begin with the same twistor function that was ). As we will see, this piece will end up vanishing for ? X ). Other than this, eq. ( n 3 tr 3.2 R Y 4 3.6 4 ( N N local ), for which the bulk field ( 2.61 ) is the spin-0 boundary-to-bulk propagator. Let’s make F Z    Y ), we get the following partition function: ( ) insertions at multiple boundary points as before, let’s focus ` ). , plus 2 for the null polarization vector 2.53 . Recall that such a choice of special boundary point defines a 3.1 ` Y iδ Y ` ( ( ` − = 1 term, corresponding to the 1-point function, which strictly = exp δ F )] = exp n ) is a function of just 4 variables, and corresponds via the Penrose ∼ ) to the local potentials as in ( Y point Y ) in terms of some convenient basis of boundary modes? Y ( ( n ( Y F F ( [ ) is defined in terms of a star-product Taylor series, and the determinant F ) an coincide with F F HS 3.6 single HS Z Z (1 + ? ) is defined as the exponent of tr field function ... Does What does the partition function The central questions for this section are now the following: At this point, we can forget altogether about the local sources ( ? 2. 1. the foundation of ourwhose discussion Penrose transform of ( correlators:the concrete the sign choice spin-0rather boundary than insertion consider instead on a 3.2 Boundary master3.2.1 field, Bulk spinor-helicity basis and itsIn interpretation this as subsection, a weHS-algebraic describe boundary partition the master function first ( of two useful bases for the argument For the full context of these questions, we refer back to section gauge-invariant rather, their gauge-invariant fieldfields strengths) corresponding from to the boundary asymptotics of the bulk least, this hasare the gauge advantage potentials, of and removingvariables the (3 gauge sum for redundancy. of thetwistor Indeed, point their function degrees the of localtransform freedom sources to is gauge-invariant that bulk of field a strengths. function of If 5 we ever wish to go back from the the relevant class of functions the local and bilocalwhich expressions we’ve been ( carrying from eq. ( consider source distributions directly in terms of the twistor function where ln det included in ( wasn’t part of the correlators ( Taylor series over Now, integrating the HS-algebraic correlators ( JHEP10(2020)127 ), ` ( ∗ ). In (3.7) (3.8) l ) into in the P ( second ) and a Y P l ( ∈ ( a F are a fixed P still makes l M origin ∈ R ) u l boundary limit ( and y ) have a number → ) L ` R ), with , u ): u bulk signature rather l Y . R ( ( 4 , or simply cospinors in − u ` L l P , and Fourier-transforms ( δ u dS ) Y R to be elements of ) will change our function `,l ` ( ( u ` ) can act as representatives C ` ), where ( l L iy δ iMY L ( u e e P Y P ) , u L . with R R into a cospinor iu ) ) u and ` u l families of twistor functions based e ( ( ( itself is the “point at infinity”. In y Y ) , or in the corresponding Poincare ). In the more standard formalism ) and + y R L ` ` `,l L ) u Y C ` iu M ( ( , u e y F − ) ( R L ). Let us now make a 4-parameter hybrid u F Y ( ( ) ). Here, the labels of the spinor parameters , u ` ` `,l ) ( R , which will serve as a choice of δ l 2.77 C y ( 1 2 u 2-parameter 2 y L ( – 24 – d − − )–( are related by complex conjugation, their product u ) 2 ) is that it constitutes a boundary limit of the free `,l ` L ( = Y d ) and their coefficients C L ( u P l R ` 2.74 R L vary over the cospinor space · Z , u . δ u u u ` i ` R 2 2 Y L . Recall that elements of and − u d d ) with coefficients L does not: shifting it along ). Thus, we can fix µ ( ) ) , u R ` ]. Let us now demonstrate how the bulk l l iu ` R ) = Y L ( ( ( `,l u e R u ( ∗ L u ) constructed from P P 46 C ` u ) on their phase defines the mode’s helicity. bulk signature, they serve as a spinor-helicity formalism for δ Z Z − P L , u ), i i conformal frame, identifies Y 4 ` R -dependent factor. This can be addressed by fixing a Y L 2.36 3 ( − − , u ( ) is just a single twistor function. To form a basis, we need a 4- u R ∗ ` defines a Poincare patch. If we consider iu R ( dS R u δ Y = e P )–( u ` ( `,l Y ( ) = ` also defines a cosmological horizon, on which the boundary-to-bulk C L ∼ δ `,l Y ` iu ( 2.35 C , normalized via ie F µ ) will have a pole. ) in the l ) by a − ` defines the 3d spatial momentum of a mode in Poincare coordinates, while ( R u P L , then 2.53 4 u refer to the handedness of the boundary-to-bulk propagators above. . The inverse transform reads: − ∈ µ L L ) ]. In particular, when u ` `γ Y u ( conformal frame on the boundary, for which conformal frame defined by ( R y 45 ` 3 3 EAdS u δ Another point of view on Putting all of the above together, we are decomposing the twistor function There is, however, a slight problem with this construction. While Now, of course, R into R Y and ∼ L ) ` R µ ( iu the dependence of bulk master field ( of HS theory, withoutprocedure embedding was space introduced in or [ spacetime-independent twistors, this limiting The basis functions of useful meanings.field In modes on thepatch [ cosmological horizon definedp by As we can see,spinor this transform decomposes they twistor intrinsic boundary terms, this makesthe them cospinors at the origin basis functions pair of boundary points, and sense as an element of e boundary point flat conformal frame defined by of the equivalence classes in at a boundary point; thesewhich are generate the respectively functions thepropagators right-handed for all and spins left-handed —of parts see these of eqs. functions, ( of boundary-to-bulk theu form the bulk, the choice of than propagator ( parameter family. We already encountered flat JHEP10(2020)127 , , 1 a − U z (3.9) (3.14) (3.15) (3.10) (3.11) (3.12) (3.13) ). In a ) becomes ) into 2.36 . . Y , 3.9 ; L ) R x u ` ) ( u L ( ` ( u y C ). Note that the ) .  2 ` ) both degenerate iy ( ` ) e x x 1 z iy zd (  e  3.11 u ( ) ) = − R along the geodesic that x P ( R ), up to the same factor u ) 1 ` ) ` u iy 1 2 b ( x  e ( + . ) 1 2 + dy , and then treat these Weyl ) ) ) , u x ` ` ) x R . a π ( ( =  x − 2 y b ( ) − `u ( ) by a nearly identical variable ` a R dy ( u ( , this becomes: z u u F 1 ), since it gets multiplied by y zl ) at . R ) ab 2 ; + ) 3.9 ) x ` x ) x ). Thus, the bulk field ( ) zl  ( `u ), if we replace the spinor arguments ( − zl L  x y − z a ( b 3.11 ( 1 ) ( 2 2 C in ( = δ y ` , u 3.8 d ( P ( P 2.19 ) ) ) R + + ` ` x F b ( ∈ u ( ( ) = b zly ) ( ` y P a ) x ( ) = (1 + dy ( ` x Z `,l y x + ) y ) ` 1 z  C ) − a ` ( 2 ( ) and (  ` ( ( ) from ( ( to d  u – 25 – π  y u L L dy y ) ) ) )  L x ` end up being opposite. The reason for this will x  ( ( z `u 2.14 , u x ( 1 2 ; y 2 ( z `u P R F R P 2 Z R ) ab L u ` iu i ( ( L/R u P iu approaches the boundary point ) = y  u `,l = 2 x 2 ), and neglect the second term in (   x C l = ` ) becomes: d ) = (  x ) ) 1 z ) ( ) ( exp ` x P b x x ( ( y and b . ( exp ( a ). In this limit, the bulk spinor spaces P iz y ∈ z ) ) as a boundary limit of , u P Z dy 2 1 + x ) ) L d 3.3 2.2 x x  → =  iz a ( ( − , u π ) ( 1 2 L/R u ) 4 R dy u u x -proportional second term in ( → ) ( ; u = z ) ( x x ) ( ( , u L x `,l ) , as in ( ( ) then becomes: u ab l C x C ) P − , u x ( ) with ) 3.9 ( x u x = ; P − )  ( x ( x ( u ): = ( ` ; P y ), in the following detailed way: ) C ( x 2 ` x ( ( ( times our spinor function d P ∈ u C P z ) as before, but now with by an extra exponential factor. It was important here that we Alternatively, instead of replacing the spinor arguments by brute force, we can perform ∈ x ) z  ` ( ( This establishes of did not neglect the and ends up contributing non-trivially. Shifting the integration variable from The r.h.s. of ( u handedness labels on become clearer in section an honest change of variables: where we used the measure definitionsjust ( We can now replacey the integration variable The integration measure Now, consider the limit in which connects it with into slight change to thedecompose notation it there, into we a renamespinors pair the as of twistor two Weyl argument separate spinors of arguments: works in our present formalism. We begin with the bulk Penrose transform ( JHEP10(2020)127 ), and . 3.20 (3.18) (3.20) (3.25) (3.21) (3.22) (3.23) (3.24) L  u ) ) has its 1 + u − R 3.23 , u n )–( u = results from the ( 1 4 u `,l , ˜ 3.22 ): C ) 0 ) ) ; (3.19) n ) ; (3.16) 3.7 , u Y Y 00 , u ; ; 0 1 u 0 ( − , . n . `,l ); ) u, v ˜ . u, u u C ); ) ( ( ( 0 0 Y Y ; (3.17) )  ) ; ; , u `,l 0 0 0 00 `,l `,l u k ) ˜ u u k C − 0  ) have an especially nice behavior u, u − ) 0 ) u 2 − 0 v + u, u u, u u, − u n ... u, ( ( ( + u ( u, − 2 u − ) 3.18 − ( ( `,l `,l `,l 2 3 0 u, u u `,l ˜ ˜ ˜ C C C ( ( − u ˜ u `,l `,l C ` ( , u 00 , k k δ `,l ` 0 2 u Y ˜ u δ C . . . d ) gets canceled here by the one from ( 2 − − u u 2 0  1 ( d ) takes the form: ) in this basis reads: d ` l 2 u , ) u + δ ) 3.6 l 2 ` l 2 ) = ) = 4 ( ˜ ( 2 ) = ) = C 3.6 u d P P 2.23 Y Y ⇐⇒ − ⇐⇒ − ) – 26 – ), ( ) Y Y u d Y/  Z l ; ; 2 Z ( ) ; 2 ( 0 0 0 0 d `,l u P , u ) ) ) C ? δ Z − l 2.54 1 v, v ( u, u u ) i u, u 4 Y Y u ( for convenience, while the factor of ( ( ) = 4 P ( ( ( i ( ), but with the propagators replaced by delta functions. ) basis, the star product becomes just a matrix product: +1 Y ), ( i − e Z 0 `,l `,l Y ⇐⇒ ; n `,l `,l 0 ( k − ˜ ? δ n C ?F 1) ? ? k ) F 2.59 ? k u, u ) 2.33 ) = ) = ) = × ) ) − ? 0 tr u, u Y ) ( Y ( Y Y ( Y )–( Y tr ( ( Y ; ( ; 0 `,l F δ ( 0 =1 ∞ u, u F k δ X n ( 2.58 ?F `,l u, u u, u N ˜ ( ) C (  `,l Y `,l ( k k F ) shows that the HS trace ( . This defines a relabeling of the basis decomposition ( )] = exp L Y u 3.20 ( ) implies that, in the ( F − [ R HS ). The slightly strange flipped sign on the final spinor argument in ( u 3.19 Z ): = Y 0 ( 3.22 so that: Note that the factor of 4( from ( own origin in HS algebra,δ being related to the star product with the twistor delta function Thus, the HS-algebraic partition function ( which is reminiscent of ( Eq. ( while eq. ( change of integration variables. Theunder basis the functions star ( product: Here, we reshuffled the factor of An alternative choice ofu variables, which is sometimes more useful, is 3.2.2 Boundary momentum basis and its star products JHEP10(2020)127 ), the ). This is 2.36 0 as the spinor u, u ( ) can be inter- 0 ) to actual bulk `,l conformal frame ˜ C Y u, u ( 3 3.22 R F )–( ) or ]. However, we’ll insist ], where null boundary L 43 49 , u 3.21 – R u 47 ( `,l . C 4) signature is complex, so are ) basis functions themselves, the 0 , , and it is there that we’d like to HS 3 bases Z S 3 u, u modes have a number of advantages. R 3 R over complex variables. Therefore, they ) or in the Penrose transform ( ) or ( ], as well as in [ L 2.22 , u 43 , R – 27 – u ), the problem merely reappears in a new guise: 34 . 0 bulk. Another is that a Euclidean boundary admits n , they do not have a well-defined support: we can conformal frame. Since it can be related back to the . Third, they provide the simplest possible explicit 4 3 ), one may consider it not as a separate construction, u, u S ( dS 7.1 `,l ˜ 3.16 C , with its Euclidean boundary and complex spinors. One ) on the Euclidean 3d boundary. As a result, the delta ) into boundary l 4 modes ( contour integrals ), ( , where both twistors and boundary spinors are real, and the , the matrix-algebra-like formulas ( 3 P 4 3.16 ) or S 3.7 L on one of the ( 6.2 EAdS ) are very formal constructs. As delta functions of a complex ), ( ) are AdS , u R HS ) and 3.7 ` u 3.20 = 2 term, where the two spinor variables can be related by complex Z 3.23 ( ( n P `,l )–( C ], but not so for general 3.19 ) for the HS-algebraic partition function 3 ). This will require a switch from modes based on an . HS 3.6 3.23 Z without its complex conjugate In the next subsection, we will describe a basis that doesn’t suffer from contour ambi- Nevertheless, this problem is not a deep one. The simplest solution is to just switch Despite such nice properties, this basis isn’t quite satisfactory for actually computing mode decompositions ( 3 3.3 Bulk master3.3.1 field, Basis, starIn products this and subsection, partitionfunction function we ( at last obtain some actual values for the HS-algebraic partition evaluate guities, and is in factR adapted to an but as a particular way of addressing the contour and convergence issues. signature to Lorentzian contour and convergence issues have beenfor studied now systematically on [ remainingreason in is our eventual interest inthe a most symmetric global conformal frame, namely spinor integrals in ( aren’t fully well-defined untilto we convergent specify integrals the for the integrationeasy desired contour, classes to and of do ensure functions for thatconjugation the it [ leads never say for which values ofattempt its to argument evaluate such a deltaanswer function is vanishes! ill-defined. As When, awith instead result, of if coefficients a we single basisjust function, like we the consider a integrals superposition in the star product ( the partition function.the spinor Just spaces asfunctions twistor in space ( invariable SO(1 The decompositions ( First, they’re more directly relatedand than boundary the field original modes. twistorcussion function of Second, spin-locality they’ll play informula an section ( important conceptual role in the dis- preted in terms ofsquare particle roots quantum of on-shell mechanics boundary onintrinsic momenta. the boundary Similar boundary, point formulas with of have been viewmomenta developed in are from e.g. used an [ directly, without recourse to3.2.3 spinors. Importance and disadvantages of the As we’ll review in section JHEP10(2020)127 ) j 3 = ) S ) Y , a x ; ) . x − x ( (3.29) (3.26) (3.27) (3.28) ( M ) − ( ) y y ) is given C x ) in this ) ( x x x − Y − m ( ( ( , where we modes with ) here is not , which lives ). at m − F im µ ( s ) e j M x x iMY ) ; . Consider, for ( + e ) − MY s µ x y x ). Recalling that iMY 2 ( ( x ) e − x decouple from each y Y ) ( δ ; x ) ) = ( x C + x x y ( ) Y − ) = ) ( by cos( x ; x C y ( Y ) − x ; ( x ( m x − ( C m ( iMY . ; , we now also understand the e C , with polarization defined by im x
, j x ) ) with ) ie − ) that these will be spanned by ) = ( y ) ). Similarly, spin- ) ( 0 Y x − x Y ; C − ; m embedding space, i.e. a bulk point ( − n 2.42 ) ) that will compose the HS-algebraic modes, parameterized by the polar- 2.41 x + ( y x ( 4 ) ) = M , Y 3 m P − x ( ( 1 ; ( C i S ), ( − x M e R ( ∈ m 0 M 2 ( − m ) d ( x 2.38 x j into a single Dirac spinor (or twistor) , but at the antipodal point k Y ) at the fixed bulk point imm − ) ( ( µ e Z ) ) Y x i x x x . We can arrange all these modes as the Taylor ? . . . ? k ; m δ ( – 28 – = j  − y x ) ( ) is, as the notation suggests, just the Penrose ) + ( y 0 s x Y m ( 2 C iMY M ; ). For the left-handed factors, we will only need the ) im ) = 1 ; ) m ie hyperboloid. The appearance of , of a single master field x x ) and Y ) M ). − ? e − x ; 4 x − ( ( 2.22 ( ( is the opposite of that at x  y ( M ) P k ) = ( C x ( y 3.13 imy x )–( ) = x Y e ? − ∈ − ; Y ( k EAdS tr ( held fixed. The expansion of a twistor function or x ) ) x m ( 2.21 x . What are the corresponding values of the master field ) ( ( x j x M j C ? δ ) ; ), evaluated not at m  conformal frame on the boundary, with residual symmetry SO(4), ) ( x x ): Y 3 ( ( m − y S ( ? One can see from eqs. ( ) iMY F 3.6 , with x x ( a ie − m MC M ) of 4 . The restriction to integer spins can be realized by considering only even . The natural modes in this frame are spin-weighted spherical harmonics d ) 4 ) = x 2.36 ) = ( Z Y − ( a Y ; frame. An ; m 3 x M ) = EAdS ( S ( , arranged into the integer-spin irreps of SO(4). Conveniently, this is precisely the + x Y 3 0 are spanned by the unbalanced monomials C Let’s now evaluate the traced products of The twistor function corresponding to the master field ( ) k ∈ S x F a ( µ The task is simple, sinceother the in right-handed the and left-handed star spinors productstar at product ( of two exponents: reversed handedness labels in ( partition function ( where the coefficienttransform function ( on the second branch ofsurprising, the due to itsthe Fourier-transform handedness relationship of ( spinors at ization twistor basis takes the form: by the inverse Penrose transform: This will be our basis of twistor functions for the expansion, in either combined the two polarization spinors m orders in the Taylor expansion; equivalently, we can replace with angular momentum at the fixed point the balanced monomials a pair of Weyls spinors > and is obtained by fixingx a timelike directionon in the structure of a linearizedexample, modes master of field the spin-0 source, which can be organized in spherical harmonics on to an JHEP10(2020)127 ) is   x  ( (3.34) (3.32) (3.31) ) x ab ( q discrete , we are P n m ) . x ]. Applying ( p 50 . m p even odd −  ) with homogene- q ) n n Y Y 1) ( ( − x f δ , k ) + ( , this is trivial: the trace a 0  ) n ) ; (3.30) x ) = ??? (3.33) m x )  ( ). As we will argue in de- − y ) x ( q ) y − ( 0 x k Y ) ( p ( m x ∂ ), vanishes. Putting everything m . . . ∂Y m ( p ) x m m − 1 x ) with homogeneity 0. Thus, we δ a m p + ˜ − m − ? Y ( p ( ) 3.34 i ( 1) y x +1 e ) with homogeneity 0 mod 4. Now, m ∂Y ( f ( 0 p − δ y  ( Y 0  ( 1) ( x ? ). The spin-parity argument is that HS +1 f =1 imm δ − p n n p tr ( − imm ? X = ) e e k q x P 3.32 =1 a ( n p ˜ + n =1 m ) = ) = ; tr p X P ) – 29 – 0 0 i x i ) ( ... m m x  y ( p   ) 1 0 must clearly vanish by rotational symmetry (note  x − − a ( m x y y exp δ p ) of this product. For even ˜ exp m ( ( ! 1) k > n    1 1 k i ? δ ? δ prefactor in ( − 3.28 ( × ) ) recursively, we arrive at: n ) subscript on all the spinors. For the right-handed factors, =0 ∞ m n k =1 X imy   x ) = /i p X ) e − x − Y 2 (corresponding, as expected, to spin 0), while the trace oper- 3.31 ; − ≡ y ) = ( n − ( ) ) subscripts were suppressed. The alternating form of the star ) y )–( x x δ ) x ), and with it the entire series ( ( ( M x ) is directly analogous to the alternation between “particle-like” and ( ) should be real, but then having it nonzero would lead to a complex Y ˜ − m m x ( ( p Y (0) singles out the component of 3.29 x ( + ˜ δ f m 3.31 x ) ? δ , this term should also be regarded as zero, due to either of two x n ( =1 ? )–( p 4 X y ) = ( i -independent exponent on the first line. On the other hand, for odd ? . . . ? k x : complex conjugation and spin-parity. In brief, the complex-conjugation argu- Y ) δ   Y ( 3.30 ? f Y ; ? tr 1 exp M ) has homogeneity × ( Y x ( k x algebra distinguishes even spins, correspondingity to 2 twistor mod functions 4,δ from odd spins,ation corresponding tr to conclude that tr that the traces areantisymmetric). symmetric This in their leavestail spinor in the section indices, zeroth-order while termsymmetries the tr spinor metric ment is that tr partition function, due to 1 The traces on the r.h.s. of order While this trace isn’t obviouslyargument. well-defined, we First, can let see us that Taylor-expand it it must vanish as: by a symmetry Let us now takeis the just HS the trace ( forced to take the trace of a delta function: where this timeproducts the ( ( “black hole-like” states in HS cosmologicalthe perturbation star theory, discussed products in ( [ the star product alternates between delta functions and exponents: where we suppressed the ( JHEP10(2020)127 , + µ . ) x x a ( must (3.36) (3.37) (3.38) (3.35) m . HS even odd = . Z )  a n n n  2 local ) M x Z M ( q , we will return ; ) and the basis 4 m x )  orthogonal to . ) x − ( p x ( 4 3.35 ( q , 1 m C m p R ) n ): x 2 − even odd ( p q ∈ M n n m 1) 4 µ ) are perfectly regular if we 3.26 p d v − − , q 2 ): Z 8 3.35 + ( 1) n/ ) 3.6 N/ 0 − ... x 1) ) . In fact, they become trivial, with ) − 2 is very different from − ( q 1 c ( + ( M m ) M ) ( HS x ; ≡ basis that remains ambiguous: the basis x basis , are present. In section x − Z − n n 3 ( q 3 ( p − S ) = ( m R M m ) } ) C x  Y )] = (1 + i 1 − mode of the form ( ; ) by 2 . This breaks the spacetime symmetry down to ( p 0 Y – 30 – x ; M +1 m M n 4 p ( 2  d 3.36 M X x = i ( q x single Z n +1 =1 2 p c k basis is that the traces ( p X [ +1 = {z )) 3 n n q factors on a HS ? . . . ? k n S Y n 1) exp sets the mode’s magnitude. We thus have a simple, explicit P ; Z ) now follows from plugging the traces ( n HS − c × Y ( =1 Z ; M n p HS equal to each other, ( ) evaluate to: =1 ∞ Z x M ) is an integral over the complex polarization spinors n X n P ( Y x M 8 ( 3.28 N k | ( exp F  ? 2 tr ? . . . ? k n/ evaluated on a finite source! ) into the general HS-algebraic formula ( ) of ) 1) Y HS − ; ( Z 3.27 3.27 1 )] = exp ( M Y ), the integral is not really necessary: we can instead consider discrete sums of the ( ( x = Y F , without a specified integration contour. There are two ways to handle this issue. k ; -dependence remaining: [ ) ( One crucial feature of the x ? M M HS − ( tr a ( Z x basis elements. Alternatively, we canSince fixed a the preferred left-handed choice of andcomplex “real” right-handed integration conjugation, contour. spinors this ofnatural requires SO(4) way some are to further neither proceedi.e. breaking real is a of nor tangent to spacetime related direction choose at by symmetry. a the spacelike bulk One point unit vector There is one final pieceexpansion in ( our treatment of the m First, since thek partition function is perfectly well-defined on individual basis elements where the scalar coefficient example of 3.3.2 Real contour and relation to the This allows us to evaluate in fact be even. For now, it is ourset first all hint the that arguments no Note that only even orders, labeledto in the ( surprising absence of the odd orders, and argue from first principles that The partition function expansion ( together, the traces ( JHEP10(2020)127 , , ) 3 ); ). x R − ( 1.3 in an 3.20 3.17 ). As (3.39) (3.41) (3.43) m 3.3.1 l ( bases is ), i.e. to )–( ; (3.42) ), ( P Y 3 ( ). Without
R x ) 3.8 ` 3.19 ( ) ; (3.40) iδ constant 3.43 modes’ matrix- m − ) and l 3 ( )–( 3 R . As promised in S . In these variables, im ) iM`lM l ( that is 3.40 ) (though still only at 3.2 ¯ ) + m σ , ) + basis function along the ` R R 3.32 u 3 , ,  u here refers to the particular ) S ) ` are all elements of − (  µ µ − 0 v L term in ( x m L u ) 0 l u − ( boundary frame itself is given by ( . The reality condition that sets ( u, u ` ) µ ) l 3 of the l ( u`u x ( im S ( iM`lM a m iM` 2 1 + and or im + bases to each other. To do this, let’s 0 M + 0 L R proportional to = 3 ) ) + `u ` ) ) + ) µ S ( `u , u ` R l l modes, as in section ( ( R L u R m u u iM`u m u 3 is equivalent to choosing a geodesic that goes im + = and + R − – 31 – ); µ L ). We recall that − L 3 µ v u u v u ( M R ( ) basis, and apply to it the transforms ( ) reads: ` ) 3.26 ( + ` iMu ) for the HS-algebraic partition function with the one 3 ( formalism can be viewed as a regularization of the iM µ S 1. + 3 basis can now be recovered using the im x 3.41 + ), as 0 im ( S − l 3 3.38 ( + R 1 2 S + )–( boundary frame. The 0 basis ( = u`u `u R ,P 3 will now make = 3 x i 2 ) L S S · ) ` u`u µ `u 3.40  x ( ` ) of the ` i iu 2 ( L P ¯  m iu ) of the exp v ). This demonstrates that HS algebra in the 3.26 . i ) exp x − 3.32 ( i 3.21 ¯ m − v ) = 4 exp ( ) = 0 L ) = 4 exp ) = , it is easy to relate the 0 , u L u, u R ( . The endpoints of this geodesic are the two boundary points: , u u u, u `,l lightcone section x ( = 0 element of our R ( ˜ C 4 3.2.3 u `,l proportional to , a `,l ( 1 ) ˜ C C Now, let us notice that a choice of x `,l M R conformal frame. From the discussion of SO(4) representation theory in section − C ( 3 S it follows that the correspondingthe twistor function must bebulk proportional point to that definesthe the 3.4 Disagreement with theLet us CFT now result compare ourobtained result by a ( standard localwe calculation will in focus the on boundary the simplest CFT.contact case As of corrections. a discussed spin-0 in In source, section where particular, the we local correlators will don’t choose require a scalar source mutually consistent. In fact, the one. The role of regularizerit, is the played modes’ by star the with products it, all the result in stareven complex products orders). delta produce functions, proper as functions, in as eqs. in ( eq. ( The star products ( product formula ( boundary spinor spaces m the basis transformations ( where we recall thata the final spinor tweak, arguments we can decompose the parameter from which wesection can construct aconsider basis an of element ( Using our already familiar techniques for manipulating spinor spaces, we obtain: proportional to through SO(3), and enables us to fix a contour by imposing a “reality condition” that sets JHEP10(2020)127 . 2 . ) = Y (3.49) (3.50) (3.51) (3.44) (3.45) (3.46) (3.47) (3.48) + 1) ; i ` j 2 ( Y `xY , it is clear (0) a   κ ) that a local Y  . However, we exp 4 ` : / π 1 1 2.52 σ 2 − − ∓ 2 σ = . = ` . 2 ) 3 1 4 ]. The partition function . m d + 1) x 1 , · ) + j . ( . The Penrose transform of ` 2 Y i ! x ) ( ) as: iY `xY  2( ` + σ is now a constant. This is just  Y δ 4 / 3  + 1) σ − S spherical harmonics with angular 1 4 2.50 exp j (0; . Z ( / σ  x 3 ) − with mass 1 8 , for which the integral is just mul- S − π x σ  2 πσ , iσ I 4 π · 1 − k N/ 2 = πσk φ 1 ∓ ` ) 2  ( σ 2 ∓ ∓ + 3 4 1 + ), where ) over the boundary point π σ =  2 2  ) = Y − ` + 1) 3.2 ( π 4 ) =  ln 3 Y ` j / h ( d δ Y ( 1 x ) ( tr – 32 – σ + 2) . Performing a Taylor expansion in ) for this argument reads: x ) = Y − − 3 ? δ j ; N Y 2 1 S ( ` ) . Altogether, the Penrose transform of our desired ) is given by eq. ( ) is described by the twistor function ( = (1 + k j ( 2 − `  3.38 x Y πiσδ Y ( π  − , and perform the integral not on the twistor functions ( ; (0) − HS ln  ` ? δ over (0) κ free scalar fields  ( ) representations of SO(4), whose dimension is ( 1 = Z iF ) 3 J ` j 2 2  S 4 3 (0) Y , N ) = 2.3.3 ( Z j 2 κ = exp ln ` -independent term Y − σ δ of  ( + 1) Y 2 ) as: 2 3 ) becomes: π j F 1 k ( ∇ Y local 4 S ( ) = Z 3.2 ∓ =0 = =1 ∞ ∞ F Y j X k X ( ) reads:  F N N ) = − − Y 3.44 (   x ? δ . These are the ( j ) = exp = exp Y should be given by the integral: ; ) (the sign ambiguity won’t matter here). Thus, the twistor function for a constant ` ( σ Y ( is given by the functional determinant ( local ` (0) Let us now fix our twistor function’s normalization. Recall from eq. ( δ Z π i iκ 4 local momentum The conformal Laplacian on these harmonics has the eigenvalues: The partition function ( Z the partition function on To evaluate this, we decompose the scalar fields into The HS-algebraic partition function ( Let’s now perform the corresponding CFT calculation, following [ tiplication by the 3-spheretwistor volume function 2 ( from which we extract It now becomes easy to integrate from SO(4) symmetry thatThus, all we terms are with left nonzero with powers must the vanish upon integration. Unfortunately, it’s unclear how tocan integrate employ a trick from section directly, but on the correspondingeach local free boundary bulk insertion master fields at insertion of the spin-0 operator source JHEP10(2020)127 : are σ ) are (3.52) (3.54) HS Z if one is 3.54 )–( and ) between local 3.53 local Z . 2.77 that set all the odd  ) 5 σ , ( 3.3.1 ! ]. The central observation O ). As we have seen, ) are not the same. To be 1 , there was one delicate step: dt + is neither even nor odd. One 2 with respect to the source in light of it. 4 2 2.31 3.52 πt σ Z ; (3.53) HS local -point correlators. Moreover, they  Z 1 cot n  Z ) 2 6 -order coefficients in ( t − σ ) and ( σ 2 ( was to carry out the superposition in- 6 π O even 1+4  3.48 , while 3.4 + √ σ 1 + 4 despite the identical correlators constitutes a Z 3 σ σ 2 – 33 – 2 8 π 2 3 HS Nπ is indeed fully captured by the correlators, with no Z − − − 2 2 , this ambiguity can be consistently resolved in either

σ σ local and   Z 2 2 8 8 = exp local Nπ Nπ Z as explicit linear superpositions of local boundary insertions = = local + 1. With a modicum of help from Mathematica, the sum can be 3 Z j HS is an even function of match, but the higher-order coefficients do not. The most obvious local Z ≡ 2 Z -point correlators HS σ k ln : somehow, the dictionary between local correlators and HS algebra fails n Z ln to zero. However, the subleading HS ). The best we could do in section Z Y ; `, λ The answer is, at first sight, not obvious. Recall that we did not construct the ba- ( ) s ( Could this be the reason behind the apparent failuresis of functions linearity? ofκ section directly, via the Penrose transform into bulk fields. How can we follow the fate of the In our construction of thethe HS-algebraic sign correlators ambiguity in in section the productonly of interested three in local insertions ( direction, each leadingsources to and a twistor functions. different However,arbitrary sign if twistor we choice functions, allow it in for may arbitrary become the superpositions, impossible dictionary to leading resolve to ( the ambiguity consistently. failure of linearity to commute with linearfailure. superpositions. We will now point out4.1 the technical core Spontaneously of symmetry this breaking as the culprit the disagreement is coming from, and how to think4 of Explanation: sign ambiguityThe and disagreement discrete between symmetries also different! We have thus establisheddiffer, that despite the having local been and constructeddiffer HS-algebraic from partition in functions the the same spin-0 sector,contact where corrections. As far ascomputing we two can different things. tell, this In disagreement the is following genuine: sections, we will try to understand where The coefficients of difference is that may be tempted toorders pin in the blame on our argument in section of this paper iscompletely concrete, that let’s the compare partition the functions Taylor series ( of ln converted into an integral: which in turn can be evaluated in terms of polylogarithms [ where we renamed JHEP10(2020)127 ), . If Y ¯ ( f spin- to be x δ g ? ? and = ¯ fails ) of twistor the discrete problematic ). Since the Y ( Y , and precisely ¯ mode function doesn’t matter f ( f ? g the symmetries . We can define f a 3 break → ), or simply set all ¯ S M ) ) for nonzero spins ) ) = 2.1.2 Y 2.61 → ( twistor functions, and ¯ Y invoke even more f 3.26 ( 2.61 a (though not always for ¯ basis. Here, we will use f ), ( 3 M spontaneous 3 ): the discrete symmetry is ), ( ), ( . We briefly mentioned the S 2.55 complex conjugation ) via 2.31 2.55 2.77 Y ( polynomial ¯ f in the → HS , to justify the vanishing of tr ), we needed to ) Z Y of twistors in section ( 3.35 ). This means that the local boundary spin- a f ). That issue deserves separate treatment, 3.3.1 ¯ Y Y ( ), but is maintained in the sense of relating the complex conjugation x 2.60 – 34 – →  a , the sign ambiguity in the star product 2.31 , δ . Their analogues ( (the anti-idempotence of ) Y 3 a Y ( ) and the zero-angular-momentum ¯ ` M δ ) is preserved by complex conjugation, except for a sign flip 2.52 ↔ ) onto the bulk and boundary spinor spaces, as well as for a correlators ( x imaginary 2.21 M  ( or are real under the twistor complex conjugation, the same is true ,P ) from ( ) are b µ ) bilocal a ` ¯ λ Y ) through this opaque process? By keeping track of symmetries! In ) ( ( µ ` ), the reader may notice that we ignore their P 3.47 γ ↔ iδ 2.31 µ λ 2.61 ∼  ), ( ) from ( Y ( 2.55 x . We already invoked these in section iδ Now, the star product ( Having laid out the general logic, we will now separately discuss each of the two In contrast, the star products in the local correlators ( Complex conjugation and spin-parity are well-known symmetries of HS algebra. They here, because of the restriction to integer spins, i.e.in the even non-commutative powers). term. As afunction result, acts the as complex conjugation a Hermitianwe conjugation symmetrize with over the respect order to of the factors, star as product: in the correlators ( for the projectors the corresponding delta functions 0 insertions ∼  and angular momentaaveraging are over also imaginary, once we impose real polarization tensors by The first of the twoanti-idempotent discrete complex symmetries conjugation is the complex conjugation ofgamma twistor matrices functions ( discrete symmetries. As welators focus ( on theirrelationship relationship with with the the localwhich HS-algebraic will corre- be given elsewhere. 4.2 Complex conjugation behind the failure of the superpositioninto the principle: finite-source as modes we attempt ofresolved to section consistently, integrate and local insertions this allowsto the be discrete recovered. symmetry between the two sign choices symmetries. To ourliterature. knowledge, Moreover, this we basic can identifyassociate observation the it has symmetry with breaking not the as sign beenviolated ambiguity in once made the we before 3-point make product a intwo sign ( the choices choice in to ( each other. This suggests that the sign ambiguity is indeed the mechanism are respected unambiguously byformally HS extend to algebra more acting general functionsfor on and the distributions. star They products cantheir be of verified HS the explicitly traces: various basis again,in functions for order from the to section arrive odd at orders a in result). ( particular, we will be interestedparity in two discrete symmetries: which implied the vanishingthese of same all symmetries to odd make orders a of more general argument. sign ambiguity ( JHEP10(2020)127 , , ). ). n n Y ( ) for , and 2.31 -point F n 3.52 ) and the ) and the correlators . Since spin ) = + a n 2.61 2.61 iY iY ( ), ( ), ( F , the r.h.s. is real, basis satisfies this → n , i.e. the distinction 3 a 2.55 S 2.55 average, namely zero. Y anti-commutators obeyed by the real , and likewise for the odd- n not ): at odd satisfy the symmetry, due to spin parity in e.g. the 2.61 at odd don’t HS , even spins correspond to functions , giving a ), ( s Z , this was indeed what we concluded 2 ). In particular, for a constant spin-0 ).  -point correlators ( 2.55 ), then the reversed multiplication order 2 n ) also obviously commutes with complex . 3.36 3.3.1 3.48 − 3.6 obtained by integrating them. In particular, )–( local 2.23 imaginary ) for the HS-algebraic correlators: at odd Z – 35 – local 3.35 Z 2.61 and ). However, in the correlators ( ), ( . In section HS 4 Z dS 2.55 ) should be with an ambiguous sign , should be imaginary. vanish 2.61 HS ), and odd spins — to functions that satisfy , this helps establish the sign ambiguity as the driving force to Z as the even function ( ). This is a strange conclusion indeed. The only way to satisfy it Y ), ( ( obtained by integrating them HS ), where the l.h.s. (when symmetrized over factor ordering) is real, 4.1 F 3.6 HS Z ), we are dealing not with commutators, but − Z 2.55 local 2.31 in ( 3.6 Z basis, as expressed in ( ) = ) is imaginary HS 3 S Z iY ( 2.31 F in the HS To sum up, by complex conjugation symmetry, the HS-algebraic correlators at odd On the other hand, this symmetry argument is clearly Putting everything together, the complex conjugation symmetry of HS algebra implies Z is encoded in twistor functions of homogeneity this is why thethat restriction to is even essential spins forpartition is unitarity function a in ( consistent truncationthe of anti-commutator HS of gravity two (and even-spin twistorto one functions a is surprising an situation, odd-spin quite function! analogous This to leads what we saw for complex conjugation. At the holomorphic level, i.e.crete symmetry without with invoking the complex same conjugation,between fate. even we This and find is odd another the spins. symmetry dis- s In of twistor space, it actsthat by substituting satisfy In HS algebra, the commutator of two even-spin functions is again an even-spin function; As discussed in section behind the disagreement between 4.3 Spin parity as well as the oddsymmetry, by part vanishing of at oddpartition orders. function The local spontaneous symmetry breaking via the sign ambiguity in the 3-point star product ( point star product ( but the r.h.s. isthe imaginary. r.h.s. of What ( enablesThe this same symmetry is breaking true for isbut the the with HS-algebraic sign an correlators ambiguous ambiguity: ( sign, which makes for a zero average. the 3-point correlators areconstant real spin-0 and source nonzero; isclearly real, similarly, visible the in but partition the isafter function formulas neither symmetrizing ( ( even over nor theshould odd. be order imaginary, of but The factors the symmetry and r.h.s. breaking is imposing is real. real This polarizations, feature the is l.h.s. clearly descended from the 3- without accepting complex correlators andand partition the functions odd is orders forfor in the odd- source, we evaluated correlators, or by the partition function is of no consequence.conjugation. The trace operation ( that the correlators ( order pieces of the factors equal, as in the partition function ( JHEP10(2020)127 is a and with 0. A 1 S , glued bulk is (0) HS B J 4 ρ > Z ∪M for AdS A µ ). However, the ρ` M , where the 2 ∼ = 3.36 S . There, the twistors , and that µ ) originates: it is the 4 n × ` )–( 1 ), where the l.h.s. (upon 2)) have a real structure. S , 2.31 AdS 3.35 0 and spacelike-separated if 2.31 to reproduce the CFT result. ) and vice versa. remains. Instead of lurking in basis, we already reached this B > 3 0 ` S M ( · local 0 ` Z i ) = in the and A HS HS ) and vice versa, and with their timelike and M Z ( Z B 1. The boundary 2+1d spacetime is given by − i − ), the culprit is spontaneous symmetry breaking , with mostly-plus signature. The – 36 – M ( µν = ) with homogeneity 0, which corresponds to spin 1, − η but not the right ones ) = 2.61 x setup in more detail. The embedding space is now I Y A · ( 4 f ), ( x M ) = ( -point correlators. In particular, both the scalar A + AdS n i 2.55 with M are timelike-separated if ( 3 have nonzero 3-point functions despite their even spin. Again, , ) should vanish for even spins and odd 0 + 2 = 0, along with the equivalence relation I R (2) `, ` ` µν 2.61 · J ∈ ` µ ∼ ), ( 3)) and boundary spinors (i.e. spinors of SO(1 x , µν obeyed by the T with 2.55 3 , 2 not ) picks out the piece of R Y ( ∈ f µ ) and in the correlators ( ? ` timelike circle. In fact, we would argue that this “default”, non-unwrapped topology 0. The boundary has the topology and conformal metric of 1 Note that our use of the embedding-space formalism is predicated on not unwrapping Again, the problem manifests at odd orders. The symmetrized star product of an odd < , whose metric we again denote by S 2.31 3 0 , ` 2 · spacelike infinities identified as the is the only onethat that we’ll compute is here consistent will with be global insensitive HS to this symmetry. issue. In any case, the quantities given by the points points pair of boundary points ` timelike circle. It can bealong viewed their as null a infinity union as of two 2+1d Minkowski spaces this signature have unambiguous signs, 5.1 Boundary topologyLet and us propagators describe theR Lorentzian we will see, thebreaking sign of ambiguity discrete in symmetries. thisalgebra. There are case However, no the indeed longer disagreement disappears, any between the hints along transition of with from linearity the violation correlators associatedthe in to HS correlators! finite sources, As we the will disagreement see, now the appears products already that in make up the HS-algebraic correlators in At this point, it’sresult worth of recalling a where Gaussian thecomplex integral sign spinors over are ambiguity avoided a in altogether complex by ( (i.e. working spinors spinor in of space. Lorentzian SO(2 InThis much provides natural of contours the for the HS twistor literature, and spinor integrals inside star products. As via the sign ambiguity: inwhich every makes case, for the a symmetry-violating r.h.s. zero has average. an ambiguous sign, 5 Conflict 2: Lorentzian boundary correlators conclusion, in the formconclusion of is the vanishing ofthe stress all tensor odd ordersthe in mismatch ( can be tracedsymmetrizing back over to the the order 3-point of starin factors) ( product is even-spin, ( but the r.h.s. is odd-spin. Again, both number of even-spin functions istrace tr again an even-spin function.and vanishes On on the even otherthe spins. hand, correlators the Thus, ( HS theeven-spin spin-parity sources symmetry should of be HS even. algebra Again, predicts for that JHEP10(2020)127 , ) 2) , are are 5.2 a (5.4) (5.5) (5.1) (5.2) (5.3) to be x U has an b a ) F µ G γ , , which now  . Instead, the   (0) . ` . In a standard . J − I , where the time ` . φ = , with the opposite ) 0 , since ` = µν ` ( 0  s η ` reads: timelike-separated points ...µ ) 1 4) signature. As a result, s 2 , µ = + ( S J where a star product leads } 3) signature. Twistors ) , ν × ` 2 ) + permutations ( 1 , γ s S µ +1 p . γ ...µ ) { 1 0 s , ` ( µ ` p ). These are now delta functions with . There is an invariant notion of real · ` A ∗ ( U ` ) 0 (timelike separation) 0 (spacelike separation) ( ) of the scalar operator 2 a 1 G ` 0 (timelike separation) 0 (spacelike separation) =0 ∞ s δ X > < U √ n ` > < =1 0 0 now squares to +1, and can be represented π Y p 2.46 3 ` ` 0 0 4 a · · d ` ` = (   · · ¯ ` ` U – 37 – n a ` ` Z ) = ¯ U 0 → + Ni 0 ` is real. We take the gamma matrices ( I 0 · a `, ` 0 ` ` = · φ ` ( 2 · ` U 1 0 ab ` 2  √ i G 2 I 1 I − π − √ 4 ¯ φ √ π ` π 4 ( 4 3 that makes global sense on d connected 1      E topology, this is not a valid inverse of ) = − Z ) 0 3). Their complex conjugation properties follow those of SO(1 n 2  , ). For boundary points, we will continue to use real “projectors” ` = ) = agree, and we can simply write: S ( 0 `, ` = ix ( F × 2 (0) `, ` S G  G CFT 1 ( × S S F 1 is a Lorentzian propagator for the fundamental fields (1 S ...J G 1 2 1 and ) G − 1 2 `  S ( ) = × , associated with real delta functions 1 = x (0) ` S 1 2 J  G ( G D P The Lorentzian CFT has the action: ) = ` ( sign from the onewe’ll that need led to to reverseto “real” gamma signs a matrices in scalar in product everynow SO(1 formula of from two vectors. section P The chiral spinor projectors at a bulk point We now turn to twistors, spinorsnow and the HS algebra spinors in of the SO(2 SO(2 spinors. The complex conjugation simply as component-wise conjugation twistors, and the twistorreal metric as well. However, their anticommutator is then where 5.2 Twistors, spinors and star products Here, we’ll avoid this subtlety by only considering correlators at Globally on the unwanted (and imaginary) extraunique singularity propagator at the antipodal point the path integralthe would Feynman be propagator. time-ordered,periodicity In making can a the be causally ignored, propagator consistent the on patch Feynman propagator the around reads: r.h.s. of ( where Lorentzian QFT (with a non-compact time axis), the correlators derived directly from It will suffice forread: us to consider correlators ( JHEP10(2020)127 . . 4 00 , πi/ ) (5.8) (5.9) (5.6) (5.7) < ` − (5.10) 0 . Since +1 ¯ p f π e , ` √ p ` < ` g ? ` is ( = ¯ G . dx 2 ) n =1 Y ix p Y − ( χ f ? g ), which leads to a ) should be positive, e 00 ` 1) R − ? δ 2.26 ) signature (for timelike- ( . 2.10 . n ) 4) signature. The HS star − ) , , Y π (0)  ( ) now comes with a natural − ` f (4 0 δ Y | 2 ` 0 0 ) ), is manifestly odd under the · − ` 0 will be convenient, enabling us 2.22 00 for ( n 2 · ` ` i . 2 5.9 · ` > − iY `` | 0 0 , while that of 00 ` ` ab 4 − ` = 4 ) · · )( `  0 ) 0 ` 0 ) = ambiguous! The value of an imaginary πi/ ` ` ` ) ` 0 · · ` +1 ( p ` π e − ` exp ` u 0 not 2 χ 0 ( · √ 2( ` `` f 1) p 2 · ( ) ` s i ) – 38 – − i ` 0 2 ) for timelike-separated points on the Lorentzian ( ` − ( 2  − u ( − ), with the Hermitian property ( = n ) = ` 2.51 ), this Gaussian integral evaluates to: i =1 +) signature, and δ ) = Y 4 n p ab , ), with the quadratic form in the exponent inherited ) ( 0 ) are now real, their integrals ( 00 2.22 Y 0 A ` Q ` ` ( ( 2.31 U ( 00 u ( ` )–( ); more specifically, since time is circular, the sign must be q ` ? δ 2 P with a positive sign. 00 , and thus the sign in ( δ d ) ? δ 00 = ) , ` ab 0 0 Y ) 2.21 `
( ( A over the real line is ) ) is completely determined by the signature of the real quadratic ` Y < ` P `, ` δ ( ), the signature cannot be mixed, as one can see from calculating Y 0 0 Z -point trace ( ) over 00 dx ` ( 5.9 2 n n , ` ` 0 ix ? δ ): it is + for (+ e 2.11 ) 00 ` < ` ` . The only possible Lorentz-invariant conclusion is that the sign depends `, ` R 0 Y ( ): ` ( ` P , i.e. well-defined distributions on the real twistor space, as opposed to the δ 5.7 ? . . . ? δ ↔ ) then reads: ) ` over Y ( 2.30 1 ab ` δ A i 1 ? By recursion, the Since the delta functions tr real argument boundary reads: distinguishing cyclic vs. anti-cyclic permutations ofWhich the of “default” the time ordering twoarbitrary leads choice to a of positive orientation“default” sign, ordering for and the which time to circle. a negative sign, For depends concreteness, on let’s our associate the form separated points ( the determinant).interchange Now, on the time ordering of ( This time, however, theGaussian sign integral on theTherefore, r.h.s. the is sign in ( from that in ( Similarly to the Euclidean case ( product ( The 3-point productGaussian can integral again ( be deduced from this via eq. ( enforced by an absolute value: Here again, our restriction toto timelike ignore separations this subtlety and proceed without writing absolute values. The basic 2-point star more formal delta functions withproduct complex is argument again from given SO(1 twistor by ( space is real,choice the of contour. integral star-product formula ( a JHEP10(2020)127 1
) ` ). , Y ; → 3.6 (5.11) n n ` ` ( ) is then )–( ) can’t be ) from the (0) alternating 3.5 5.10 5.2 either along a 5.11 → · · · → +1 ?. . .?κ p ) 2 ` as future-pointing, ` + permutations Y 1 ;
` ) 1 → ` the discrete symmetries 1 Y ( ` ; → compensating n (0) when summed over permu- ` n κ ( ` ? (0) respect ) to the next one vanishes p ) mean that, unlike in the Euclidean 5.10 ` ) ? . . . ? κ ) that represent local spin-0 insertions ) Y 5.10 ; be respected, since in the absence of sign Y 5.10 ` ; is the number of propagators that end up ( 1 ` ). However, then the r.h.s. of ( χ ( (0) κ must (0) – 39 – 5.11 reproduce correctly the correlators ( κ ), unlike in the Euclidean. Second, the Lorentzian ? , then Y 1 tr ( ` and the last propagator ` can’t δ 2  ` -point generalization ( ∼  ). The sign is again not arbitrary, but can be determined as → n ) 1 5.5 Y ` ; ` connected ( E , the HS-algebraic trace ( ) (0) n n κ ` ) and its ( ), the twistor functions 5.9 (0) 5.2 + 1 is the winding number of the cyclic sequence χ ...J ) 1 ), the HS-algebraic traces ` is the propagator ( ( G (0) , where 2.55 χ J To summarize, the HS-algebraic “Feynman diagrams” tr On the other hand, the alternating signs in ( Several conclusions follow. First, we see that to reproduce the reality properties of D 1) − In the previous twoand sections, HS we algebra ascribed tothis the certain explanation disagreement sign is between issues. ratherdisagreement? the While anti-climactic. boundary perhaps In CFT satisfactory Isdeals this at there with section, the some off-shell technical we boundary more level, will particles, conceptual while present reason HS for such algebra the a sees reason: on-shell ones. the We CFT begin in path integral linearities. However, theirthe signs correlators are from different the from CFT those path integral. required to correctly6 reproduce Explanation 2: on-shell vs. off-shell boundary particles CFT correlator. One couldsigns in into principle the introduce sum by overconsidered hand permutations as some in the ( expansion of an HS-algebraic partition function,(for such spin as 0, in ( atdiscrete symmetries, timelike-separated and points) integrating are over them no does longer not sign-ambiguous, appear to no threaten longer with break any non- In particular, for odd tations. This is in agreement with the discrete symmetries, but in obvious conflict with the expect any hidden non-linearities when integrating over the correlators’case insertion ( points. CFT path integral: 3-point product ( that were spontaneously brokenextra by sign flips their incurred Euclidean uponordering. counterparts. reordering the In star-product This fact, factors, follows theambiguities, due discrete from there to symmetries the the is changed no time mechanism that might break them. By the same token, we don’t and for every intermediate propagatornot choose to a cross future-pointing thepast-pointing. or time past-pointing coordinate arc of so as the correlators ( must be chosen real follows. First, note thatfuture-pointing we arc, can or get along its from complementary each( past-pointing point arc. The sign inaround ( the time circle,draw choosing the the future-pointing first arc propagator at each step. Alternatively, if we where JHEP10(2020)127 ) ) ) ) ` 2 ˆ ( ` , ` 0 s , ` 0 (6.1) ` ) over ...µ ( ) ). The 1 δ s , ` ` µ ( ) 0 ( even just 2 ` s A ˆ ` F?...?F ...µ `, ) ( 1 ( s ? ( µ δ J ) already span the ) = ), maps to the star 2 ˆ ` , ` 0 , 0 ` 2 = 6 spacetime coor- ). The first is given in ), which for the spin-0 2.45 ( ` ( 2 ) contains only physical × δ 3.6 3.5 ) Y 1 ( ˆ ` F `, ( ) of 3 . ) and Π δ ) , ` ) 1 0 ˆ 2 ` ` ` ˆ ( ` , ). However, the sources 0 , ` σ ). ` 1 ( ( ) ˆ ` Y σ δ ( ; ) of 4 twistor components. This makes Y ) ˆ ` ; 1 G ( Y ˆ ` ` ( ( = ) is the propagator ( `, (0) F 0 ) (with spin-0 sources, to simplify the dis- ( 2 (0) κ δ Π `, ` 3.2 ` κ ( seem natural and unavoidable: the underlying G 3 ) = – 40 – 1 ) = d G . ), which describes a local spin-0 source at a point Y ˆ ` , ` ( F 0 , Z local 0 ? ` gauge-redundant, due to the field equations satisfied F ` ( Z tr ( , we will review the identification between HS algebra 1 δ 1 4 ) = ) ˆ , where ` − Y ). The second is given in terms of traces tr 6.2 2 and ( more ). Thus, an agreement of the partition functions ` `, Π ( ( F , ` δ φ 0 G HS ` 1 Z ), such that: . ) = Y within the algebra of infinite-dimensional “matrices” Π( ( 5 , ` 0 n ) maps to , to understand the strange agreement-up-to-signs of the Lorentzian F ` ) . ) are even G 2 , ` 6.3 already calls for a complete isomorphism between the two algebras. 0 , then their products Π σG ` 2 ?F ˆ ` 1 F and with a degree-of-freedom counting argument, which applies to any spacetime 1 0 are gauge-redundant, due to the conservation of the currents ˆ ` sources Π( 6.1 , maps to the twistor function ˆ ` product s > The matrix product Π The function Π( The trace tr(Π Now, let’s understand the origin of the dimensional mismatch itself. One way to do However, such an isomorphism is clearly impossible, by simple degree-of-freedom count- 2. 1. 3. this is in termsdegrees of of gauge freedom. redundancy. Sowith does The the twistor local function spin-0bilocal source for spin-0 sources ing. The bilocal matrixdinates, algebra while consists HS of algebra consists functionsthe of Π( disagreement functions between algebras simply have different dimensions. Note that the focus onconsider spin pairs 0 of does spin-0 not insertionsat restrict Π points the isomorphism’s generality.entire Indeed, algebra if of we bilocals Π( In more detail, theand would-be twistor isomorphism functions is between bilocal boundary functions Π( the space of boundarywithin fields HS algebra.isomorphism between The the would-be two equality algebras,case via reads of the simply: the linear partition mapping ( functions is a statement of correlators from section 6.1 Dimensionality ofSuppose algebras that the CFTcussion) partition in fact function agreed ( withterms the of HS-algebraic traces partition tr( function ( signature. We will thenmore explicitly focus in terms on of a on-shella vs. Lorentzian complexified off-shell boundary, sense). (in where Euclidean, In on-shell theand section particles issue exist the only can algebra in be ofapply framed operators it, on in section the space of on-shell boundary particle states. We then section JHEP10(2020)127 , . ) ) ,d ,d Y 2 2 ( 6.2 R R (6.4) (6.2) (6.3) F = 3, a d . Every is viewed ]; we will ,d 3 2 R -dimensional local d ]. A simplified Z operator algebra ) conformal sym- 52 , , d . In a flat conformal 51 , ,d 2 25 R in through the origin in , the energy-momentum of a , via: µν a ,d 2 L Y R ), the HS algebra elements , ` ] ∈ -dimensional Lorentzian spacetime. ( σ d ] φ ν µ ∞ . ` L boundary particle states. Let us review ) Y. µ ρ [ [ ∞ ) come into play, even if initially we were ] so as to match the identification of ( ν ( µν δ 3 ` , ` ). And, as we’ve seen, when 0 ` ` was quite abstract. In this section, we will µν Y γ ( totally null plane = 4 I L 1 8 } on-shell φ – 41 – ) 0 = ρσ = 2 ` 2.2.1 ). ( ` µ I µν ,L ( p ¯ φ L σ µν L { are fixed up to normalization by the SO(2 of a free massless particle in the 2+1d boundary spacetime. reads: µν is simply the square of a twistor L µν generate the conformal group. Now, in our case of interest µν L dimensions becomes a L µν d defined by a “point at infinity” L 1 2 in − -dimensional spacetime then becomes a lightray through the origin in d = ∞ ` lightray · ` ) are operators on the space of off-shell fields There is another, equivalent way of understanding the mismatch, which can be made , ` 0 ` implying that the totally null bivector The Poisson brackets of metry. The normalization can inas turn a be translation fixed generator. [ The resulting Poisson brackets read: the particle’s phase space consistsframe of totally null bivectors particle described by the particle’s energy.spacetime Since the in theory an is embedding-spacepoint conformal, in formalism, we the as can thewhile represent projective a the lightcone in Combining this with the “magnitude” that encoded the particle’s energy, we conclude that 6.2.1 Twistor space asConsider a phase free space massless particleWhat in is a the conformally classical flat phasea space lightray, of namely such the a particle? particle’s A wordline, point along in with the an phase space affine consists “magnitude” of to represent bring into play ain more the concrete quantum point mechanics This identification of has view: been madeversion, HS from adapted algebra various to points a is ofnow 2+1d just view summarize boundary in the it and [ here. utilizing twistors, was presented in [ can be viewed as operatorsthis on the identification space in of detail. 6.2 HS algebra asOur quantum definition mechanics of HS algebra in section algebraically, all the degrees ofonly freedom interested in in Π( the spin-0 source particularly concrete inΠ( Lorentzian boundary signature. In the same way that bilocals by each “leg” of the bilocal operator JHEP10(2020)127 ) : ` can 6.5 (6.8) (6.5) (6.6) (6.7) ˆ f , as can 3) sym- is just a a , ) Y , with the l ( a ) represents . Thus, HS y Y Y ( δ only up to sign. a , constructed from . in the HS-algebraic Y Y 4 . These spinors have 1 4 a d . 7.1.3 Y ). The result reads: of HS algebra )) ) Y 6.4 ) defines ( ). It is the same factor of 4 as is again fixed by SO(2 2.22 6.4 ? δ ( a 6.6 ) – Y ) Y . ( , f ) l . ( . General quantum operators ab 2.21 ( ? ) in the conformal frame defined by ( I ab ab tr ` y I iI µ 6.2 1 4 1 2 = 2 γ ) l − o ( b y is then represented by the Moyal star product ) = ) of the phase space coordinates = – 42 – ] = 2 1 4 ˆ g Y Y ,Y b ˆ ( ( ab f a ). From the point of view of the boundary particle, ˆ Y = f ) that make up the twistor Ω Y , l µ Y f a ( n 4 p 2.25 ˆ ) represents an operator that flips the sign of Y P d ). As we saw, the twistor delta function Y ) ) is not quite identical to the quantum-mechanical trace ∈ l ( Z ( δ Y ) y l 1 4 ( . Such a splitting is precisely given by a choice of two bound- ( ( represents the totally symmetrized product of the operators f y ψ ab = ? I n ˆ a f ∼ tr ). ) and ab ` ...Y is a momentum spinor. Pure states of the boundary particle can be ( 1 3.6 ) a P l . The role of configuration and momentum variables is then played by ( l Y y ∈ ) which is precisely the star product ` ). The operator product ( and n y a ), ` ˆ Y Y ( ), and can be thought of as an explanation for the factor of ? g ,..., The twistor delta function The trace operation tr ) 1 3.22 , but is closely related, via: , and the one constructed from the symplectic form ( a ˆ Y f ( ˆ ab Y square root of the on-shell energy-momentum ( In other words, expressed as wavefunctions splitting twistor space intothe two symplectic 2d form Ω subspaces thatary are points Lagrangian, i.e.the totally spinors null under a simple meaning within the boundary particle’s mechanics. In particular, However, the issue is more subtle, and we’ll6.2.2 return to it Boundary in spinor section Having space interpreted as twistor configuration space space posed as into a configuration phase variables space, and one their may conjugate wonder momenta. if This it would can necessitate be decom- in ( partition function ( be evidenced by its adjointone action might ( expect this sign flip to be invisible, since eq. ( Here, the factor of 4I is due to the ratio between our twistor measure convention that ( f algebra is just the algebra of operators in the quantumtr mechanics of the boundary particle. Now, consider the quantization ofis this particle upgraded mechanics, into wherebe the a Poisson represented bracket commutator as ( [ ordinary functions metry, while the normalization can be fixed by matching to ( implying a symplectic form: The Poisson bracket for this new phase space variable JHEP10(2020)127 ) 0 ), ). Y and ( `, ` 6.7 state ) ba- . Of a ( 0 0 , then F δ i ) u Y l a u ( | y u, u boundary = ( ) = ) describes 0 + ) `,l Y l ) fields, and is ( ˜ . Specifically, ( ` C `, ` a ( ` i y = 0. In other ( ` δ y 0 ) ) of a boundary G | l ( u =  ( y on-shell and a . In particular, the ψ should disagree with Y u off-shell HS ) of the twistor metric, = Z 3.2.2 ) l ( 2.20 y ) in the flat conformal frame 6.8 = 0, which describe states of the : as a phase space function ), as we can see from the overall ) ˆ l F φ ( y  − ) that describes a local spin-0 insertion ( ψ Y is just the state with ) is totally null under the twistor metric, ( ` − ` i ( δ ` ) is just the Wigner-Weyl transform between 0 P 0 | ∼ that solve the field equation ) ) is just a matrix element between the – 43 – 0 onto a particular quantum state 1 u, u Y ( | − ; ` ` `,l u, u  0 ( ), but to ˜ ( ) C l ih and decompose twistor space as (0 = ( ` `,l can be thought of as a determinant (or an exponenti- κ l y 0 ˜ = 0. This is particularly explicit in the C | ) between the basis states G 0 − ( φ HS ) and ψ Z  u, u Y . The two are similar superficially, in that they both calculate ( ( ), and . One difference is that, in Euclidean, the operator represented `,l F l ) l ( ˜ a local ( C projector y P Z , we restrict for simplicity to spin 0. . On a Lorentzian boundary, this minus is suppressed, since the real ∈ ) not to was written in the context of a Euclidean boundary. However, the ) 5 ) l l ( u ( , where the Hilbert space is realized as functions y y ). 2 ( ) l ). As a result, in Euclidean, there would be an overall minus sign in ( d ) between ( ψ ) 3.2.2 ` y . Recall that the 2d subspace ( are canonical conjugates, and 3.2.2 ` state. − y -point correlators. Let us now explain this “miracle” from the point of view of . 3.25 ( 2 3.16 ) . It is only natural that determinants over such different spaces will disagree! ` i n l 0 d ( ψ 1 4 u y | ) sends − → ) Y ) = ( and l Consider the twistor function What begins to seem strange at this point is that HS algebra ever managed to reproduce To sum up, we see that In secret, we already encountered this formalism, in section ( δ ) y Y ` ( 4 ( (up to normalization) a if we fix a secondy boundary point words, it is thedefined state by with vanishing energy-momentum ( the correlators match toabout. the As extent in that section they do, and howat the the sign point disagreementswhich come makes it,submanifold from of the the phase point space. of This implies view that the of phase boundary space function particle mechanics, a Lagrangian in fact composed ofwith propagators source the CFT’s boundary particle mechanics. We will work in Lorentzian signature, and explain both how The above understanding of HS algebrathe makes CFT it even path clearer integral that determinants over boundaryfields, fields. i.e. solutions However, tofree HS the boundary source-free algebra particle. field sees equation In only contrast, the CFT path integral is over sis of section momentum-spinor and the 6.3 On-shell HS algebra vs. off-shell CFT correlators d integration contours impose positivity on all the measures andated delta trace) functions. over thethe boundary boundary particle’s field Hilbert space, equation i.e. over the space of solutions to course, section Euclidean signature changes little, except to complexifyconfiguration the variables phase-space coordinates by sign in eq. ( This extra sign arises from the minus in the decomposition ( ψ transform ( two representations of a quantum-mechanical operator vs. as matrix elements a sign-reversal operator on the phase space. Acting on pure states, this operator sends JHEP10(2020)127 . ) i ` HS +1 0 Z | p , we (6.9) (6.11) (6.10) of the , ` p 7.1 ` ( , local G ) are given ) depend on i Z 1 ` 6.9 6.9 0 | n ) of the state ` 0 0 ` ( ` is to the future or to i h = 0 that is symmetric 0 n ψ ` ) takes the form: ` φ 0 | 1  − ) is ill-defined, since there’s 5.10 0 n ` ` , despite the matching of the ( 0 ` h on the r.h.s. of ( HS ψ . The solution is: 3) case), and despite the absence i Z ` , ... . i +1 a half-integer. Therefore, not only p 3 1 4 , ` ` . This is in fact a general property of 0 0 ω ) 0 0 | | − ` 2 p ` ` 0 0 `, ` , then the conformal Laplacian becomes: h ( j and i h + 1) G 2 . ` j ` | ( ` 0 | j 0 , making 1 ∼  – 44 – ). ` 1 2 ih − ) 0 ` 0 2 0 ` + | ( ω 5.10 ` j boundary, the sign of topology. If we denote the frequency along the time ∼ h ψ = 2 2 ), which reproduce the off-shell propagators = S
S  ) | HS × can be thought of as a wavefunction in the spin-0 sector. We then analyzed the causes of this ω × Y 6.10 ), and the sign depends on whether 1 | Z i ( 1 ` S n 0 S ` 5.5 | , and with it lose the power of manifest global HS symmetry, 0 local ` . If we restrict again for simplicity to timelike separations, then Z HS 0 ` h Z angular momentum by 2 ? . . . ? δ S , and with it the connection to the local boundary field theory. We , we will address our core motivations, and discuss how choosing ) Y ( = 0 on the local on-shell solution will have a sign inconsistency upon traversing the time 7.2 -point correlators (up to sign, in the SO(2 1 ` Z n φ δ only in its squared form  i . Globally on the ? any ` ` , and the = 0 translates into 0 tr | ω is the propagator ( φ  G ), but The agreement up to sign of the HS-algebraic “Feynman diagrams” with their CFT Thus, the l.h.s. of the HS-algebraic “Feynman diagrams” ( 0 ` ( ` or give up on will now advocate forwill the discuss second replacing boundary choice, locality andThen, with explore in boundary section its spin-locality as implications.can a guiding benefit In principle. the section project of higher-spin dS/CFT. In this paper,boundary we vector identified model a and disagreementcorresponding the between HS-algebraic expression the partitionof function contact corrections to disagreement. After all is saidmust and either done, give the up disagreement on must be accepted as a fact. We that we encountered on the r.h.s. of ( 7 Aftermath: choosing sign issue won’t bother us,the because state the HS-algebraic “Feynman diagrams” ( counterparts is now clear.by the The on-shell inner wavefunctions ( products up to time-ordering-dependent signs. These are exactly the time-ordering-dependent signs Thus, ψ circle (of course, the inconsistency disappears if the time circle is decompactified). This the past of no globally consistent time orderingsolutions between to circle by there’s exactly one such solution in the neighborhood of where where the inner product What does this wavefunction lookaround like? the It chosen must point be a solution to JHEP10(2020)127 n c (7.1) (7.2) at each ] reveal n 20 a – was deduced 18 HS boundary limit, is madness. Isn’t Z ) in terms of bulk → , Y ( -point correlators? local   F n ) for . Z  ) } ) } 3.6 Y Y ( ( over HS ) to change the coefficients Z {z {z factors 7.2 factors ?...?F ?...?F n n ) ) Y Y ( ( F | F |  n ? a tr =2 ∞ n X n – 45 – c =1 ∞ ) + X n Y ) for the CFT correlators. But if we are content with   ( is not as crazy as one might think. But then we must F at each order. However, this leads us straight into the 2.61 ] for the bulk theory. Just like the bulk fields, the argument HS n , such as the string length. In the bulk c Z 20 −→ )] = exp ) Y Y ( ) with: ( boundary limit, a massless propagator does not become pointlike. F [ F 3.6 → HS Z ) into essentially anything! The theory has thus become empty: having cut our ties ) of the partition function can be subjected to non-linear redefinitions. As long as these Perhaps, then, choosing Also, let us recall how boundary locality comes about in AdS/CFT, from the bulk 7.1 Y ( Clearly, there is enoughin ( freedom in thewith the redefinitions local ( CFT and placed our trust in HS symmetry alone, we are left with a partition are trivial at first order,perturbations, they or won’t affect in the terms basicpossible of interpretation redefinitions, boundary of modes. it still Whileorder: leaves HS us symmetry with severely the restricts the freedom of a single coefficient with an arbitrary coefficient impasse that was outlined in [ F the CFT correlators in thevalue first HS place? symmetry higher The time thanwe has agreement must come with admit to the the face CFT,means this and most replacing inconsistency. we general eq. If cannot partition ( we have function both, compatible then with HS symmetry. This The usual argument for boundary locality is no longerface obvious the at consequences all! of leavingfrom the the CFT HS-algebraic behind. formula ( a Our formula partition ( function that doesn’t match the CFT one, then what’s the point of matching this finite non-locality lengththeory. becomes But, scaled as we down mentioned to inremains the zero, Introduction, non-local resulting recent results at in have shown all athe that scales! HS local bulk gravity interactions boundary In to particular, berescaling the as of non-local quartic-vertex the as results bulk a of massless [ propagator. Even under the infinite not the highest principle?the Is boundary there vector any model actual at evidence finite that sources, the as bulk opposedpoint of HS to view. just gravity matches the The bulkconfined quantum to gravity theory some is non-local, length but scale its non-locality is usually 7.1.1 Leaving theFrom local the CFT point of behind viewthe of CFT AdS/CFT always orthodoxy, right? choosing Doesn’ttheory? it Isn’t provide disagreement the with veryanswer it definition with synonymous of counter-questions with the of having bulk our made quantum own. a gravity mistake? Is symmetry Yet we — can in this case, HS symmetry — 7.1 Boundary spin-locality JHEP10(2020)127 ). 0 : (7.4) L u, u ). This , u u ( R u ψ ], i.e. a set approaches , to reduce ϕ [ x `,l C O ). This doesn’t 0 , as ) of two boundary ) in the role of an , we find that only L L u, u ϕ ( , u ˜ , u . C 3.2.1 R ) spinors to their linear ] R , the limiting behavior of ; (7.3) u L l u ( ] ( , ϕ ), and restrict ourselves to , u C `,l ,ϕ C u ) from first principles, up to [ C ( C R [ L ) with u ψ 3.6 L L u , u 2 , with d R spin-local ϕ u ) is equivalent to locality in ( S R ( L − u C 2 , u d ϕ e ) almost completely. Disregarding trivial R ) l D ( u , ϕ P ˜ , global HS symmetry means simply that we C Z Z ( L – 46 – 6.2 ] = )] = , replacing L , ϕ ). As we discussed in section ] is to impose spin-locality as a requirement on the ) C and [ , u x . On the boundary, we can take a much simpler route, , apart from what can be reshuffled arbitrarily by field ( 12 a R – 3.2.2 . u , y Z 9 ( ) L x 3.2.2 C ) is described by the function u ] should contain no more than one integral over [ ) − ( x spin-local ( y , ϕ S ; C and , y [ x ) ( x R C − spin-local u ( ) (in the following, we will omit the subscripts on Z y along the geodesic defined by a second point l no information ≡ ; ( ). ` ) ) from section x spin-local 0 P L ( Y S ; C , u ∈ x u, u R ]. Let us rein in the freedom of redefinitions by imposing on the boundary ( ], over some set of dynamical variables u L C ] is a spin-local “Lagrangian” that contains only quantities evaluated at the 12 ) as a matrix over the space of spinor functions 0 – , u , ϕ 9 , ϕ C R [ C u [ u, u ( L ˜ C To this requirement of spin-locality we now add the requirement of global HS symmetry. In the bulk, the project of [ In the bulk, spin-locality means locality with respect to the spinor arguments of the spin-local Now, as we recall fromtreat sections matrix-algebra-like products that are invariantrestriction under fixes linear the transformations of spin-localterms Lagrangian that are either constant or linear in the dynamical variables Here, single point ( For this purpose, itcombinations will ( be convenient toaffect switch the from the spin-locality ( condition: locality in ( We will demand that theS partition function be expressedexternal as source the that path couples integralof of to operators an a from action complete which set allthen of others simply can that “single-trace” be operators constructed. The requirement of spin-locality is to the spinor variables unfolded field equations,respect which to are the auxiliary obtained variables constructed from by analogy Vasiliev’s with the equations CFT by path integral solving and its with standard relation to bulk fields. master field the boundary point the master field spinors clutter). Therefore, the boundary version of spin-locality should be locality with respect not the standard spacetime localityallow of us the to CFT, reproduce but thesome spin-locality. HS-algebraic choices As partition of we function signs will ( claim and see, any reality this strict will conditions equivalence between that bulkwe we spin-locality will will and now the need introduce. boundary to spin-locality However, fix it that by should hand. be clear We that do one not is inspired by the other. redefinitions. 7.1.2 A spin-localTo path rise integral from thisthe despair, bulk let [ us recall the emerging answer to the analogous situation in function that carries JHEP10(2020)127 ) 0 ), u ( . (7.8) (7.7) (7.6) J u, u ) ( 0 ϕ ). The ˜ C u ( IJ functions 7.5 J g of the sec- . ϕ N J ) ) ), depending 0 0 I ) = u N β by appropriate ). The integral u ( ( I could have been I J u, u I ( δ ϕ 7.7 ) to which ϕ ˜ ϕ 0 ) C 0 u ) ( u ) exhausts all possible I 0 ( to be symmetric if the u, u I ϕ ) or Sp(2 alone, and ) ; (7.5) ( ) 0 ϕ ˜ C IJ u N 0 u u u, u g ( ) ( , ( u I (2 ˜ u J 2 C ( ϕ in the Lagrangian ( N ϕ O I ) of ∓ i J u d ϕ ), ) u 2 I  0 0 ] ( d β u ˜ I N C 2 ϕ ) over the action ( Z u, u ( u d J ˜ and 7.3 C , i.e. the set of operators to which they 2 I ) is now “even more complete” than is 0 d J J β ] in the theory. This can only be true if I I ϕ det[1 + Z β [ α u, u  ) + ( O ˜ u C ): ( ) + ) + 0 complete J . This still leaves the coefficient u )] = u I – 47 – ( 0 ϕ I ϕ ) 2.47 − ϕ u ) u, u u ( ( ( I u ˜ δ ( C [ I of spinor functions J u ϕ I ) must be 2 0 α N u ϕ d h 2 ) d Z u, u u ( independent, we may linearly relate them as . However, the added requirement of HS symmetry implies spin-local ( J 0 . We take the internal-space metric ˜ Z . This leaves us with a spin-local action quite analogous to the C I I I Z u ˜ ) to even spins. As with the local CFT, we will denote the range C N α ϕ ϕ 0 2 ] = -independent term can be replaced by the identity are linearly related, and on whether they commute or anti-commute. I are possible: ˜ ] = ] = and C u, u ... I and must then be proportional to the identity, and can be normalized to I I ( , ϕ ϕ ˜ ϕ I u C I J , ϕ , ϕ ϕ I I I , ϕ β ˜ of the and C , ϕ , ϕ [ ˜ ˜ I J alone. So far, these variables can be either commuting or anti-commuting; C C [ [ I ϕ 0 L α u ), with some number spin-local 7.5 ) of S 0 spin-local in this case as 1 Let’s now address the constant coefficients We are now ready to perform the path integral ( u S commute, and anti-symmetric if they anti-commute. I ( I I -independent linear transformations of is Gaussian, leading to: As an aside, weusual note in that AdS/CFT. the set Normally, thestructed of theory by sources multiplying would together also the includecouples. “single-trace” “multi-trace” operators operators, However, con- suchthat “multi-trace” would operators be both cannot spin-local beexternal and coupled sources HS-invariant. in In to this the a sense, theory. source in a way (bilocal form of) the free vector model ( couple should generate allwe possible impose operators a symmetry onon the whether internal-space indices: U( The coefficient the identity by rescaling coefficient u ond term arbitrary. To fixour it, set we of invoke external one sources of the principles we borrowed from AdS/CFT: ϕ as we will see,Instead the of correct having choice willwhich be will the restrict opposite ofof the usual one forϕ the CFT fields. From the point offunctions view of of spin-locality both alone,that the the dynamical only variables non-trivialform (i.e. ( higher than linear) terms in the Lagrangian must have the terms quadratic in JHEP10(2020)127 . ) ), ), 0 u ( ! 7.9 ψ (7.9) ) ), i.e. 1 u, u (7.11) (7.10) of any ( twistor ˜ i C , u . 0 7.10 n u . This last | u  ). A subtle ˆ ( ) f )–( ˜ | 1 C ). As we saw u u ) h 3.23 7.8 n − , 3.16 , u n imaginary 1 u ! − ( n ˜ , as we’ve done to go n C ˜ 0 C u ), as was derived origi- ) u ( n tr ˜ C n anti-commuting , u 3.23 as given in ( 1 and ... , and + for anti-commuting. 1) n − modes on the boundary, one I ) u n − HS n 3 ϕ 3 ( u u Z ( R 2 , u ) can be bridged by recalling the ˜ from ( C 2 =1 ∞ X n u . ( HS ... 3.23 ˜ . . . d C ) N Z ) 1 ¯ ) u 3  2 u , 0 in ( 2

, u u d , u 2 (¯ 1 ) ˜ HS l u C ( u ( Z ( ˜ P − C ) that corresponds to it via ( ˜ C Z ) = exp Y . By the same logic, we should require our spin- . Similarly, the matrix elements n 2 0 – 48 – n ( u u  ) = u ] 0 2 , u F 1) ) and ˜ n 1 C discussed above. ]. − u ], we falsely assumed the opposite, which in turn led ( ) between the twistor/spinor variables and the state ( u, u 7.9 3 and ˜ . . . d ( 3 , C for commuting variables , HS ˜ ), we get: 1 2 C u =1 6.8 2 ∞ Z × u in ( X n − 2 7.8 d ), ( . We can then flip the sign of the last spinor in eq. ( N tr ln[1 + ) to be even in their spinor argument, making the sources 0 ) in the spin-local path integral are 0 l u ( u  N 6.4 ϕ ( P I  ∓ Z spin-local ϕ  ), which translates into: n and Z Y 1) u ( n , this is a good time to recall again with humility that we had that − F ( HS ) and − )] = exp 0 Z u =1 ( )] = exp ∞ 0 X n I , the correlators for real CFT sources are reproduced by ) = ϕ u, u 3 N ( Y – u, u ˜ ( C  ( should be even in both [ 2 ¯ ˜ C F ˆ

[ f ) even in both 0 One final awkwardness remains. When working with Another discrete choice that must be made by hand is the reality condition on spin-local = exp u, u Z spin-local ( ˜ Z Again, in the original versionsus of to [ the wrong overall sign in ln can indeed restrict to functions that are even in the spinors or, equivalently, on the twistor function in sections functions nally from the CFT correlators,provided provided that that the we variables choose therequirement lower is sign a in ( bitoverall strange, sign but in not ln obviouslysign problematic. wrong in Since our it previous originates works from [ the This agrees with the HS-algebraic partition function operator local “fields” C bringing it into the form: 7.1.3 Integer spins,The antipodal difference symmetry between and realitydouble-cover conditions relationships ( of the boundary particle.should be Since even the in boundary the momentum-spinor particle has spin 0, its wavefunction This is almost identical to thedifference HS-algebraic still effective remains, action which we will now address. Expanding the determinant in ( where the sign in the exponent is JHEP10(2020)127 from upon (7.13) (7.12) ) to a modes, ). This ] of the HS 3 Y 56 Z ( 2.28 S x − is equivalent modes, since δ 0 3 u ]. Indeed, recall S 2 and u . ] ) . ` ( ) s ) describing Y ...µ ( ) even in does not obviously break ) 0 1 ` s ) for spin 0, are odd under this ( µ δ 3.26 x A − · [ u, u ) into the left-handed ( ` ˜ Y ( C ( ]. However, with the global / ) = ). This is indeed the case for the local local x 3 δ Y Z Y harmonics, this antipodal symmetry is ( ( ]: the AdS/CFT duality between type-A δ 3 ? δ S ] = 42 ) ) ` ( spherical harmonics, e.g. the zeroth harmonic ), having ) vector model s as the Hartle-Hawking wavefunction [ Y ) corresponds in the bulk to the antipodal map 3 ( – 49 – 4 ` N Y S ...µ δ ) ( 1 s 3.25 (2 δ dS ( µ O ] interpret the partition function of the CFT on the A ) = [ 42 Y ( ), as one can see from applying the identity ( ` ? δ 2.77 ) and the Y 4 ( ) turns the right-handed ), ( δ Hartle-Hawking Y ( AdS Ψ δ 2.52 the case for the twistor functions ( under star-multiplication by ) assumed a definite antipodal symmetry (which makes sense from the ); such was also our approach in [ ], the authors of [ : not ` ], and thus a crucial theoretical laboratory for quantum gravity in de Sitter 55 odd 7.10 7.10 54 , , but when analytically continued into the antipodal branch, they have a pole at 4 ) that star-multiplication by 1), are neither even nor odd. In particular, they are regular on one branch of the 53 ) to ( ) to ( . This is yet another spontaneous breaking of a discrete symmetry, which is quite . The boundary-to-bulk propagators, e.g. 1 − 0 ) being 0 x 7.9 2.42 7.9 x x Y EAdS · ( − quantum HS gravity: Following [ To summarize, we managed to recover the HS-algebraic partition function x F 4 ( → − = / future (or past) conformal boundarydS of is due to ahigher-spin remarkable gravity observation in made inchanging [ the sign ofdS/CFT [ the cosmological constant.space (though, This of course, provides HS a gravity rare is not working the model realistic of gravity of GR). 7.2 Implications for7.2.1 dS/CFT A globalFinally, maximum we for come the to Hartle-Hawkinggreat our wavefunction open own problems motivation in forconstant. theoretical the physics study Higher-spin is of gravity quantum turns higher-spin gravity theory. with out positive to One cosmological be of especially the relevant to this problem. This from ( point of view of boundary particlewhereas mechanics, and in is our satisfied bysacrificed main local in boundary application, favor insertions), of i.e. regularity in in the the Euclidean bulk. general in AdS/CFT withpropagators massless aren’t antipodally bulk symmetric fields in (for the first massive place). fields,first the principles, boundary-to-bulk using spin-localitythe and CFT global correlators HS only for symmetry.and the We for choice the were between reality forced commuting condition and on to the anti-commuting appeal sources. variables, to The weak point in our argument is that the step x map. However, their superpositions1 into bulk x However, this is star-multiplication by mismatch can be seen equivalentlyfrom from ( the point of view of bulk fields [ this becomes impossible. As weto saw in ( boundary insertions ( boundary point from ( JHEP10(2020)127 , . 4 Q  ) do dS ` (7.14) ` Q ( 3 ]. The I d Q 58 R e , the CFT commuting 3 ). S . )] = 7.14 ` ( ], and the answer at zero, as can be  I 1 ] was to change the Q ... 42 ! = 0 is overshadowed by . Their answer was the + ). This serves to flip the 4 dt σ ) given by the CFT path region with the unwanted σ N 3.4 2 πt σ  minimum + Π) to be negative-definite. 1 7.13 ] gets many things right. In ] evaluated the effective action  cot − 1 57 2 2 t 6 π σ ) to Sp(2 ], and studied further in [ . Functional integrals over  N 57 1+4 + (2 √ 3 1 O σ Z 3 2 , but the large- σ integrand 8 − Nπ 2 – 50 –

σ  ) from commuting to anti-commuting, and accord- ) by an alternative construction based on ` 2 ` ( ( , as we reviewed in section 8 I I 3 φ φ Nπ ] = exp S σ − [ values of order 1. This is a disaster that can’t be brushed , the signs of even-spin and odd-spin 2-point functions end on  4 σ σ local dS Z to = exp ] = 4 σ [ (or, more generally for bilocal sources, AdS σ ), they postulate a straightforward wavefunction Ψ[ ) of the boundary vector model has a local ` ( + ), but with an overall sign flip in the exponent, as discussed above: ], the local maximum of this partition function at I 3.1 1 . ] argued that it’s not enough to have a Hartle-Hawking wavefunction: the φ  57 ) is reproduced for small enough 3.52 ). Instead of the Hartle-Hawking wavefunction ( ` ( Hartle-Hawking I 7.14 sources Ψ Q If one accepts these changes in the rules of AdS/CFT (in particular, in the role of Such a new foundation has been proposed in [ So far, then, we can arrange a partition function that is locally peaked on empty particular, when computing propabilitiesresult ( for the constantadditional spin-0 maxima is mode now on the excluded. operator This isIn because fact, the construction for effectively better restricts or worse, just like the bilocal approach to the CFT, the construction which would normally be theappear, path but integral’s with a differentfor interpretation: operators, they which arise correspond whenCFT’s (via computing a expectation dictionary values involving the shadow transform) tothe the boundary path integral), then the construction of [ Hilbert space in whichdefined, this with Hartle-Hawking HS symmetry state taken is intoCFT account. meant and For its to this anti-commuting purpose, fields live they must replacefields the itself boundary be carefully integral over a series of higheraside maxima by at flipping some overall sign.dS/CFT, A one new that foundation would is produce apparently required a for Hartle-Hawking higher-spin state differentauthors from of ( [ As plotted in [ turned out to befor negative. a In constant spin-0 particular, source one the in authors eq. of ( [ sign of the 2-pointfunction function, the and desired indeed local ofcontinuation maximum. the from entire The effective restriction action, toup giving even being the spins opposite, partition is so essential it’s here: impossible upon to give theBut correct is sign this to maximum both a at global once. one? This question was investigated in [ empty de Sitter space, i.e.tion at function vanishing ( sources.seen A from problem the immediately sign arises: ofvector the its model’s parti- 2-point fundamental function. fields Theingly solution change proposed the in internal [ symmetry group from For this interpretation to make sense, the partition function had better be peaked on JHEP10(2020)127 , to ] on 6.2 (7.16) (7.15) ). We 57 , 7.14 55 , 42 on-shell solutions ). All that remains is . ), or those that preserve ` 8 3.48 ( φ N/ . ) 2 4 σ despite being derived from the as an improved foundation for 1 2 N/ π HS )]) local Z Y , and the internal symmetry group Z , namely the HS-algebraic partition ( (1 + ] as a guiding principle is the group of F 1 ) = 0 to be a local maximum necessitates local on a constant spin-0 source, and check if 57 ] = Y Z ), this wavefunction is globally peaked at ( σ signature, in eq. ( [1 + [ HS ] is worth remarking on: in both cases, the , this requires us to change the dynamical ? F 4 Z HS 7.14 57 commuting Z – 51 – 7.1 (det differs from to EAdS ] = HS σ is more conservative in some ways, and more radical ): [ )] = Z Y 3.6 HS ( Z F [ universe at its future conformal boundary. Aside from any HS 4 Z . In contrast, actual HS algebra, as we reviewed in section in eq. (  dS , the requirement for Hartle-Hawking HS ). Let’s now evaluate anti-commuting Ψ Z local N Z (2 O ) from u . We’ve analyzed how ( ) to I HS N ϕ linear transformations of the off-shell boundary fields Z all ] is crucially concerned with matrix algebra on the space of off-shell boundary fields. The similarity to the construction of [ We contend that the present paper provides an alternative resolution to the problem = 0. 57 = 0, as desired. φ 7.2.2 A roadAs towards with physics our interest on in cosmological HSagenda. theory horizons itself, There our is interest something inthe higher-spin conceptually dS/CFT wavefunction unsatisfactory has of about a particular the the focus of [ in others. We giveshell, up boundary HS locality, symmetry. but In gain(linearized) addition, manifest bulk we invariance fields under retain the andmaking the true, external standard that on- sources AdS/CFT path relationship in integralWe between a thus spin-local propose boundary rather the path thanhigher-spin HS-algebraic integral, dS/CFT. local. partition at function the The cost tradeoff of seems worthwhile. σ boundary CFT with anticommuting fields istion replaced with by a new, more commuting symmetry-drivendifferences; dynamical construc- our variables on proposal the to boundary. use However, there are also to flip the sign in the exponent, which gives: Aside from being obviously simpler than ( In the spin-localvariables construction of section from Sp(2 this produces a better-behavedalready Hartle-Hawking performed wavefunction this than calculation the for one in ( CFT. Just as with flipping the sign of ln of the Hartle-Hawking wavefunction’s maxima.an alternative In the to precedingfunction the sections, CFT we’ve explored partitionsame function correlators. We’ve shown howrequirements it can of be HS constructed symmetry from first and principles, spin-locality, through bypassing the altogether the spacetime-local In particular, the “HSeither symmetry” promoted in [ the conformal Laplacian is the much smaller algebra of linear transformations on the space of of [ JHEP10(2020)127 , these 7.1 . It is thus l and and 3.2 provides a better- ` HS Z approaches the endpoint x ) as Y ; x ( C ), remain important. However, we then ) of the spin-local formalism! Moreover, 0 7.14 u, u ( `,l ˜ . But, as we’ve seen in sections – 52 – C l ] — we try to take seriously the causal structure ) or 59 L , 3 , u ] implies an inflation-type scenario, in which the de Sitter ), which permits any use, distribution and reproduction in R u ( 57 , `,l C 55 , ), and its absence in ( horizon. 42 -point correlators at separated boundary points are no longer of any 4 CC-BY 4.0 7.16 n dS ], the same variables have an especially nice interpretation in terms of This article is distributed under the terms of the Creative Commons 45 . We still take as our starting point the CFT on the unobservable Euclidean 4 , delineated by a pair of cosmological horizons (the so-called “static patch”). The dS 4 dS To conclude, the HS-algebraic, spin-local partition function In this context, the switch from boundary locality to spin-locality becomes especially In our own work — see e.g. [ from the direction of the other endpoint Pohang. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. We are grateful toSudip Eugene Ghosh Skvortsov, for Mikhail discussions, Vasiliev,exchanges. Per and This Sundell, work to Mirian was Dionysiosoretical supported Tsulaia Anninos Physics by and Units the and of Quantum Charlotte theversity Gravity Sleight (OIST). and Okinawa Institute for YN’s Mathematical of & thinking email Science The- workshop was “Higher and substantially spin Technology informed Graduate gravity by — Uni- talks chaotic, and conformal discussions and algebraic at aspects” the at APCTP in boundary into an observable Lorentzian bulk patch. Let’s get to work. Acknowledgments field modes on the behaved Hartle-Hawking wavefunction at thethermore, conformal the boundary variables of in de whichmost Sitter spin-locality space. natural and Fur- variables HS for symmetry are taking manifest higher-spin are dS/CFT also from the the unobservable Euclidean natural to use variables thatcandidate live at would be one (or the both) limit` of of the these bulk endpoints. master An field especiallyare natural precisely the variables as discussed in [ natural. On one hand, sincepoint. the boundary Similarly, is the not observable, localityinterest: on it only is the quite full beside partition the does function intersect the matters. boundary On at thehorizons a other pair (or, hand, of equivalently, the points of observable — patch the her boundary worldline). endpoints of the Let observer’s us denote these by boundary, and its interpretationglobal as maximum of a ( Hartle-Hawkingaim wavefunction. to take Therefore, a e.g. nextbulk the step, and extractdaunting the difficulty Lorentzian of physics this in task an makes observable manifest patch HS of symmetry the essential. Thus, the approach of [ phase is merely temporary, and itsin would-be the conformal past future boundary lightcone becomes of contained post-inflationary observers. of pure other possible concerns, the future conformal boundary of de Sitter space is unobservable! JHEP10(2020)127 ]. Int. J. (2015) ] ] , SPIRE (2015) ] IN 06 ]. 11 ][ ]. Phys. Lett. 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