JHEP10(2017)130 Springer June 20, 2017 : October 6, 2017 October 19, 2017 : : Received Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP10(2017)130 [email protected] , b . 3 1705.06713 and Per Sundell The Authors. a c Higher Spin Gravity, Higher Spin Symmetry, Black Holes

We construct an infinite-dimensional space of solutions to Vasiliev’s equations , [email protected] Republica 220, Santiago de Chile E-mail: NSR Physics Department, G.via Marconi University, Plinio 44, Rome, Italy Departamento de Ciencias Universidad F´ısicas, Andres Bello, b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: and black hole modes arehole related backreaction by already at a the twistor second spaceon order Fourier of the transform, classical existence resulting perturbation of in theory. a aand We speculate fine-tuned black directly branch related of to modulia the space possible quasi-local that interpretation deformed is of Fronsdal free theory. the from higher Finally, black we spin hole comment black modes on hole solutions as black-hole microstates. of Fock spaceequipped operators, with corresponding positive definite toan sesquilinear scalar AdS and particle vacuum bilinear gauge forms, and function,spacetime respectively. points, static the Switching which twistor on black makes space it hole possible connectionarise to asymptotically, becomes modes, reach by analytic Vasiliev’s another gauge, large at in transformation generic which given Fronsdal here fields at first order. The particle particle modes over staticgauge higher transformation spin of black ansimple holes. all-order gauge, perturbatively in Each defined which solution particular theholomorphic, solution is spacetime and given connection obtained all in vanishes, by local the a dependence a degrees twistor of of large space freedom connection the is are spacetime encoded zero-forms. into the The residual latter twistor are space expanded over two dual spaces Abstract: in four dimensions that are asymptotic to AdS spacetime and superpose massless scalar Carlo Iazeolla 4D higher spin blackfluctuations holes with nonlinear scalar JHEP10(2017)130 66 27 24 61 30 35 47 42 29 39 9 57 37 44 3 Z 60 – i – -gauge 64 L 27 background 7 4 51 42 AdS 9 17 -gauge and Kruskal-like gauge 9 62 L 39 21 61 48 21 15 1 1 5.2.1 Weyl zero-form 5.2.2 Internal5.2.3 connection in Internal connection in Kruskal-like gauge C.1 Problem setting C.2 Integral equation C.3 Solution usingC.4 symbol calculus Non-trivial vacuum connections on 5.3 Vasiliev gauge 4.5 Embedding into an associative algebra 5.1 Gauge functions 5.2 Master fields in 4.1 Particle and4.2 black hole modes Regular presentation4.3 of generalized projectors Explicit form4.4 of internal solution Black-hole in backreaction holomorphic from gauge scalar particle modes 2.2 Master fields 2.3 Equations of motion 3.1 Gauge functions 3.2 Separation of variables in twistor space and holomorphicity 1.3 Summary of1.4 our new results Plan of the paper 2.1 Correspondence space 1.1 Motivations 1.2 Black holes or black-hole microstates? A Spinor conventions and B Properties of inner KleinC operators Deformed oscillators with delta function Klein operator 6 Conclusions and outlook 5 Spacetime dependence of the master fields 4 Internal solution space with scalar field and black hole modes 3 Solution method 2 Vasiliev’s four-dimensional bosonic models Contents 1 Introduction JHEP10(2017)130 ]. 97 ], to ] for – 95 59 95 – a priori 92 , 84 , ]) and hologra- 26 30 – , 24 13 , 9 – 7 , 4 ]), without relying on any 42 ] and references therein; see also [ 58 – ] for more recent works on the holographic 43 – 1 – 113 ] (for recent advances, see [ – 23 – 109 21 ] (for reviews, focusing on various aspects of the subject, 8 – 4 ] (for a different approach, see also [ 41 – ]), on the other hand, provide a fully nonlinear classical theory based on dif- 27 , 14 – 21 9 It is therefore highly non-trivial to establish whether and how the deformed Fronsdal Vasiliev’s equations [ One of the outstanding problems is to connect two different approaches to higher deformed Fronsdal equations on thebe spacetime directly submanifold compared [ withthis those procedure of introduces the ambiguities concerning quasi-localnoncommutative the geometry deforrmed embedding and Fronsdal of the theory. spacetime internalthe into Although quasi-locality gauge the requirement fixing, full provides one a may guiding principle; entertain for the recent idea progress, that see [ theory and Vasiliev’s formulation arethis connected end, beyond one the may linearizedto envisage approximation. make two quite contact To different directlylevel routes, at of depending the on-shell on actions level subject whether ofternal to one degrees the boundary expects of spacetime conditions. freedom action, of Thefields, Vasiliev’s or, former integrable without path, more system any whereby are indirectly, the proper viewed at in- essentially dynamics the as of auxiliary their own, involves a reduction down to a set of direct products ofThis a formalism commuting encodes the spacetime deformationCartan-integrable manifold of curvature constraints, the and at Fronsdal the internal equations expense, on-shellquasi-locality, symplectic however, and into of manifolds. manifestly of blurring the introducing spacetime additional moduli entering via the internal connection. field theory duals, thein deformed the Fronsdal absence theory of has any so fully far nonlinear been completion of ofsee the limited [ Noether use, procedure. however, ferential graded algebras defined on noncommutative Poisson manifolds, given locally by Fronsdal action following the Noether— procedure supplemented as by weak to localityspacetime conditions ensure with well-defined proper amplitudes boundaryfar built been conditions. yielding using holographic Up Green’s correlationfield functions to functions theories. corresponding quartic to in As unitary order, anti-de free for this conformal exploring Sitter approach the has moduli so space of the theory, including more nontrivial spin gravity that areFronsdal theory. presently The pursued: lattera approach Vasiliev’s (see theory review [ and and thereconstruction more of quasi-local references, bulk deformed and vertices) [ provides a perturbatively defined deformation of the free whose consistency requires special matterThe sectors and resulting non-vanishing cosmological framework constant. thustheory provides in a anti-de natural Sitter platformphy spacetime for [ [ studying tensionlessassumptions string on dualities between strongformal and field weak theory coupling side. on the worldsheet or the con- 1.1 Motivations Higher spin gravities arespin greater extensions than of two, ordinary basedto highly gravity on constrained the theories interactions gauge by governed by principle. massless non-abelian Remarkably, higher fields these spin with assumptions symmetry lead algebras, 1 Introduction JHEP10(2017)130 ]; see 91 , 90 , ] by massless ] for solutions fully nonlinear 2 , 89 67 ]. 1 , 107 66 , 3 – 1 systematic ], but the present lack of a stringy gen- ] that their higher-spin counterparts are 91 , 2 , 3 1 – 1 – 2 – The resulting configurations describe nonlinear 1 ], which requires a choice of topology and a suitable class of functions 102 ] for exact solutions in three spacetime dimensions, and [ , 17 – 101 , 15 64 , 63 In this paper, we shall extend the higher spin black hole systems of [ The study of exact solutions of higher-spin gravity is relevant for addressing other im- To pursue the comparison, the access to explicit classical solution spaces of Vasiliev’s A corollary of our analysis is that it is possible to map Weyl-ordered master fields on-shell to normal- 1 completion of linear combinations of black-hole-likeitself and equally particle well modes, to the and construction therefore of lends solution spaces consisting of general time-dependent ordered dittos obeyingembedding the of central the Vasiliev on-mass-shell system theorem into the in Frobenius-Chern-Simons the model on-shell scalar [ field sector, as required for the scalar particle and anti-particlepropagating modes. scalar fluctuation fields onbackgrounds. static spherically-symmetric Although higher spin thesesimplifications, black restrictions hole they do on not theas imply boundary any the loss conditions solution of lead method generality to developed concerning in technical our main this conclusions, paper facilitates the infinite-dimensional. We are inclined toas interpret an this indication fact, that together suchquantum solutions with black are other holes, candidates properties, rather for then thedetail higher-spin microscopic throughout black description this holes, of paper. as hairy black-hole-like we Clearly, solutions one shall should expects comment help on that in in the answering some more study of of the fluctuations key over physical such questions above. Vasiliev equations have been constructed ineralization [ of geometry has soinstance far by limited establishing our whether capability they to possess assessis a their horizon, physical precise whether or nature their — curvature not, singularity for ity etc. are finite-dimensional, Moreover, it whereas was the observed in Type-D [ solution spaces in ordinary grav- portant open questions: forstringy” instance, extension as of the (super)gravity theoryit theories, is a admits supposed natural black to question describe holeframed to an solutions, in investigate “extremely and its is whether language, howwith and, the an possibly, main infinite resolved problems by tower that the of their coupling massless physics of fields poses the of are gravitational higher fields spin. Black-hole-like solutions of the anti-de Sitter solutions with specialhigher higher spin spin black Weyl holes curvatures:also and instanton-like [ solutions, other generalizedobtained Type-D in solutions axial [ gauge in twistor space. Fronsdal theory. Note thatand this Green’s functions. does not require any equivalence atequations the may level provide of usefulof vertices guidance, whether moduli to space, extractSo or a far, to quasi-local the Fronsdal compute branch classical on-shell moduli actions space on has the been full explored noncommutative geometry. in the direction of asymptotically Vasiliev’s equations from a globally definedmanifold Lagrangian [ density on the fullforming noncommutative a differentialconjugation. graded algebra Making equipped contactpoint with between in a the moduli two trace space approaches leading operation then to and an consists a on-shell of action hermitian seeking that up agrees a with that of the deformed Following the second route, one instead seeks to implement the action principle by deriving JHEP10(2017)130 ) is a β ]. How- , v α 68 u where ( ) δ v γ ] for developments. In v β u 91 α , ( is the spin-two charge, which is 2 . In particular, black-hole Weyl , fu 2 1 A = M are eigenspinors of the (self-dual part of αβγδ α C v and α ). See appendix 0 u C is a complex function (which remains invariant under ∈ – 3 – f λ , with α = 1, and v 1 α in the asymptotic region, where − v 3 λ α − u r → 2 α v ∼ M f and α ] for the first instance of such a solution, and [ λu 3 → α see [ u 2 Moreover, as we shall comment more in detail later on, the study in this paper already The spin-two Weyl tensor is of Petrov Type D, that is, 2 spin-frame, normalized such that redefinitions tensors are distinguished, inthe) gravity, Killing by two-form the of factD a that solution, time-like Killing the vector. function real In in the the case electric of case a and static purely and imaginary spherically-symmetric in Type the magnetic case. ular, all Weyl tensors sharethe the latter; Killing symmetries, Petrov typegravity, such D a and form principal of the spinors Weylever, of tensor in is higher a spin local gravity hallmark this of identification a is black-hole subtler: solution [ so far there is no known higher-spin work, though we shall remark on these1.2 topics towards the end Black of holes theThe or paper. solutions black-hole to microstates? four-dimensionalblack higher holes spin are gravity distinct thatinclude by and are generalize a the referred spin-2 tower to Weyl of tensor as of electric an higher AdS or Schwarzschild spin black magnetic hole. Weyl In tensors partic- of all spins that the quasi-local deformed Fronsdal theory,Vasiliev’s one gauge may beyond ask the whethermodes linearized the level at ambiguity can higher residing be orders in Vasiliev’s exploited in theory. to perturbation We cancel shall theory, out not asnor the pursue its to black this analog fine-tune hole refined in to boundary the a value alternative quasi-local problem perturbative branch in scheme of this employed paper, by Vasiliev in his original form black holes modescan already at possibly second be ordereach seen of perturbative as classical order. perturbation a The theory manifestationgauge special — transformation of internal which to gauge the we Vasiliev’s non-localitylinearized start (internal) of level. from gauge, the is in connected Thus, Vasiliev which by aiming equations Fronsdal a a at fields large arise direct at correspondence the between the Vasiliev theory and theory is generated by applying homotopy contractorsThus, to the initial issue data of given whether by such the elements. self-interacting system admits particles any requires perturbative expansion a schemeexists careful describing a analysis vertex that as interchanges in particleof the and the black fully hole present non-linear modes. paper theory Indeed, will there the show exact that, solutions at least in certain gauges, particle modes interact to which is indeed atheories. necessary condition for At it therequires to classical be that a level, the candidate this resultexpanded bulk in requires of dual terms of sufficiently two of free massless the localsystem, conformal same particles which vertices; type is modes of in a mode entering set particular,the functions. of a particle it On zero-curvature modes cubic constraints the are on other vertex encoded a hand, can into higher-dimensional in specific be fibered the fiber Vasiliev space, functions, and the classical perturbation leave for a future work. offers some results that mayspin be relevant gravity. to the Indeed, comparisonperturbation the of theory the quasi-local in two which deformed approaches the to Fronsdal fundamental higher theory massless particles by are construction stable builds to any a order — black hole modes superposed with general massless particle modes of any spin, which we JHEP10(2017)130 ], 103 , ] for the 102 103 ]). 88 ] that by changing ordering 1 ] and either a conserved set 107 , 102 , 19 , ], that we think of as solutions that 15 73 – 69 – 4 – 4 appears in the black-hole solutions as the parameter of a delta r — remains an open problem. Although there exist a large number preferred in the absence of any metric-like action principle (though 3 ] a structure group was found that yields a unique complex cohomology element for a priori 88 ]. But unlike the delta function on a commutative space, which is singular thought of as 1 ] form an infinite-dimensional unitarizable higher-spin module of states (the norm 2 , Reasoning physically, we find it plausible to entertain the idea that entropic horizons On the other hand, as we shall describe more in detail below, the solutions found The global AdS radial coordinate The conservative notion of an entropy, namely as a density of states at the saddle point of a path 1 4 3 goes to zero [ cohomologies in degrees two anddefinition four of have generalized been asymptotic constructed charges. using different methods. Seesequence: also away [ from ther origin one has smoothan Gaussian element functions, in approaching a a ringas Dirac of a delta sections, symbol function the for as delta an function element in of noncommutative a twistors space star is product smooth algebra. thought Indeed, of one can show [ integral, requires theof further macroscopic notions charges of atfluctuations an infinity, or, and on-shell alternatively, Hawking action a radiation. conjugatethe [ (temperature) Alternatively, on-shell and variable action more relatedfour; as to formally, one thermal elements indeed, may in in think theevery [ of strictly on-shell the positive de even entropy degree and Rham (and cohomology vanishing cohomology in for spacetime every positive form odd degree degree). two In and [ an extension of spacetimeoscillator by algebra: a noncommutative in twistorthe space these deformed and oscillator terms, algebra satisfying remains agiven a well-defined by sign deformed there, a even that distribution though in the the twistor deformation origin space. is is not a special point is that that screen thefields higher extend spin into hairshole-like the for solutions asymptotic the in region asymptotic Vasiliev’sdespite in observer, theory the the while fact can that higher unbroken be eachtensors phase. spin thought individual and Weyl auxiliary of gauge tensor fields Moreover, as blows enter up the the being at Vasiliev black- smooth equations this packed at point. into the master Indeed, fields all origin living Weyl on investigate the details of this proposal in a futureemerge work. in broken phases ofin higher the spin gravity, unbroken whereas phase thespin as microstates symmetries are a would directly introduce spectrum visible a of mass regular scale that solutions. could give Indeed, rise the to Yukawa-like potentials breaking of higher spin gravity as gravitysolutions coupled as to higher-spin higher analogsare spin degenerate of gauge with fuzzballs the fields, [ asymptotic AdS-Schwarzschild it black region, hole is but to tempting thatthe a to contain black (macroscopic) view observer hole “hairs” such in geometry that the as modify to the create strongly a coupled smooth region solution of without a horizon. We hope to calibrations of areas may select special metrics related toin brane [ actions [ being given by the ordinary supertrace),black-hole each asymptotics one giving but rise to thatrather a suggest solution is an that has possibly interpretation identical in non-singular terms of and black-hole horizon-free, microstates. In which fact, would viewing higher and whether there exists any invariant notioncould of be an event attached horizon to — it andof whether an formally entropy defined metrics inmix higher different spin spins gravity, as as well (non-abelian)none as of higher numbers them of spin is derivatives symmetries (in units of the cosmological constant) invariant quantity ensuring that the singularity of the individual Weyl tensors is physical, JHEP10(2017)130 , with real or ], or expressing 92 , ]. 84 , ] though they must be 91 , 3 9 , 87 7 , , 4 85 , twisted projectors 67 , 66 , 9 , 1 ]), we have used the latter method to con- – 5 – 91 , 2 ). Conceptually speaking, the equations are analogs 2 ]. , that we shall refer to as | 87 2 , ] (see also [ 1 1 n/ ih− 2 n/ 2 matter fields), and all their covariant derivatives on-shell at some point / ∼ | n e P In particular, the spherically-symmetric solutions are expanded over “skew-diagonal” In a previous work [ The key to Vasiliev’s fully nonlinear theory is the fact that both the higher spin algebra In order to stress this point, let us recall a few basic facts of the Vasiliev equations projectors prescription (from Weyl ordering tosolve normal the ordering) equations in one this can basisthat map and one the then delta move would back function have todeformation. to the obtained a original In by ordering regular this — solving element, sense generating directly we the same can the solution say deformed that oscillator the algebra black-hole-like solutions with are the actually distributional smooth. distinguished by different Killing symmetries,the while all full share Weyl the zero-form Kerr-SchildWeyl coincides property, zero-form i.e., with can be the assigned linearizedinterference a one terms linear (that (in space, can while a be theswitched special twistor removed on gauge). space as eventually connection well in Thus, contains in going the certain to gauges Vasiliev’s [ gauge). oscillators constructed using algebraic methods [ struct families of exactbased generalized on Petrov factorization Type D ofinto generalized solutions, noncommutative projectors encompassed fiber and by deformed and an oscillators, base Ansatz respectively. twistor The various coordinates, families absorbed are two-form curvature in twistor space.and The a resulting covariantly system constant thus zero-formin describes in a twistor spacetime flat space. coupled connection toperturbatively This a to set system deformed of set can of deformed be Fronsdalthe oscillators fields treated fields on spacetime using in [ a two gauge dual function fashions, (i.e. by a either large gauge reducing transformation) it and a set of deformed and its zero-form modulegiven arise by as sets subspaces within ofthe one functions and connection on the to a samenon-commutative non-commutative the associative twistor twistor algebra, Weyl fiber base zero-form, space.two-forms space, Vasiliev whose In extended supporting star order products spacetime remarkable with to closed with the couple Weyl and an zero-form serve additional twisted-central as nontrivial sources for the representation. Its Lorentz covariantspin-0 basis and define spin-1 the generalizedof Weyl the tensors base (including manifold,other while states its of compact the basis theory describes [ particles, black holes, solitons and integrable system, i.e. a set ofshell curvature in constraints that terms can of beshares (large) used plenty gauge to of express functions features all and with fieldsmany topological zero-form on fields, field integration theory, making constants. the it formalism Although capableenter makes it via use of infinite-dimensional of describing spaces infinitely systems of with integrationhigher constants, local spin constituting degrees a Lie of module algebra, for freedom, the known which as the Weyl zero-form module, or as the twisted-adjoint (leaving further details forof section the constraints onsense the that, super-torsion and rather super-Riemannframe than field, tensor working a in Lorentz directly supergravity, connectionintroduced in with and together the an Fronsdal with infinite a fields, tower corresponding of the tower higher of formalism spin zero-forms, analogues. altogether employs forming These a a are Cartan JHEP10(2017)130 n n ν ν for s n s n s , the n M M M n ) = P free Fronsdal ν denotes the ], the physical ( s 3 i s as to activate 2 ) = charges 0 ν M n ( n/ s , | ν s 0 n ν M n,n δ = = s , its Lorentz covariant field n M n ν . In order to use Vasiliev’s s ]. Conversely, certain boundary M ]. Thus, for each distinct 87 2 , , 1 18 is finite-dimensional, and does not lend 5 is a non-zero integer, and n 2. These solution spaces form real higher-spin – 6 – is a Kerr-Schild vector. As stressed in [ n/ µ k , where it is possible to construct a solution of the spin- n field carrying a charge ν s s is non-invertible, which makes sense as the Lorentz spin is , where s n s µ ]; in particular, one may choose 2 M , . . . k 1 1 [ µ k ]. We would like to stress that the class of functions making up } r n M ν . { s = 102 ], for every spin , s , all proportional to the single deformation parameter 95 114 ...µ – n 1 ). On the other hand, starting from a given Fronsdal field of a fixed Lorentz µ n 92 ϕ 3 as “higher spin hairs” that needs to be fine-tuned to form a microstate. Moreover, The aforementioned questions are tied to the issue of in which classes of functions Clearly, to settle the issues of whether the singularity of the Weyl tensors is physical Conversely, one may consider deforming a whole tower of asymptotically defined (free) Looking more carefully at the classical solution for a given As found in [ 5 > superselection sector of the theory; for examples, see [ equations as higher spin gauge fieldsadmit corresponding such to a the formType-D in higher sector a spin of suitable black spin gauge. hole Weyl It tensors is described for this above reason indeed that we refer to such fields as to the linearized on twistor spaceadmissible the classes master must form fields differential gradedoperations are associative algebras [ allowed with to well-defined trace the take value. twistor fiber Onbase space general manifold, has which grounds, we an the expect impact to on be the dual to boundary a behaviour set of of observables, the thereby determining fields a on the detailed study of the propagationrequired. of As small already fluctuations mentioned, overof such the the a black-hole-like interactions study solutions induced by is is higher quiteof spin challenging, the symmetry, which due standard requires to geometric a the tools proper generalization non-locality used in the case of gravity. s it appears that the matrix not expected to be a good observable in the(not strongly a coupled gauge artifact), region. and whether or not these solutions possess an event horizon, a more Fronsdal fields, with eachtheory spin- for this purpose,for one some has ensemble toa tune single the microstate. charges In so this that sense, one may think of the asymptotic spin- spin, the corresponding linearizeditself Type-D sector to any sensibleinteractions interpretation of the as Vasiliev a systemmaking create quantum “coherent” up mechanical states, state twisted consisting of space. projectorsstates all in and Lorentz However, a spins, thus the real facilitating vector space a with physical a interpretation Euclidean in norm. terms of content contains an infinite tower of Weylstatic tensors of and all spins of that generalized arecenter; spherically Petrov symmetric, Type in D, particular, each intensor. of the which spin-two has The sector adepending Weyl curvature one on tensors singularity recognizes at carry the its (no asymptotic AdS-Schwarzschild sum Weyl charges, on that can be electric or magnetic supersingleton states with AdSrepresentations energy with positive definiteclassical bilinear solution forms lends itself [ tothan be a interpreted as Lorentzian a one microstate as of is a the Euclidean case theory, rather for the particle states; see also the Conclusions. imaginary expansion coefficients JHEP10(2017)130 , | ) 0 j ;( 1 2 + 0 j ih ) j ;( 1 2 + j | and twisted projectors bosonic model require a | 2 . 0 j n/ ), and contribute to the harmonic + 0 ih , belong to supersingleton and j j 2 , respectively, and we have sup- ( n = minimal ⊗ n/ ,... s ) j 2 ∼ |  , n 1 P  7 = n , i – 7 – 2 n/ projectors | n , which can be described by a vacuum gauge function. In four 3 , have compact spins in ( The superposition of scalar particle and black hole modes AdS ,... 6 1 2 , = 0 ], the black hole solutions to the j 1 , respectively, where | 2 and spin 1 2 n/ ]. In the latter context, the analysis of the scalar propagator on higher spin black hole + 65 j ih− 2 n/ ] by adding nonlinear time-dependent scalar field modes. To this end, we use mathe- The technical detail responsible for the aforementioned phenomenon is the presence of In what follows, we shall examine how to superpose particle modes over black hole More generally, by the Flato-Fronsdal theorem, the operators contained in The gauge function method in four dimensions differ from that employed in three-dimensional higher ∼ | 1 7 6 n e backgrounds is simplified by thedimensions fact is that locally any blackdimensions, equivalent hole to however, (in the fact, any Weylto vacuum tensors appear. solution) deform in three the spacetime gauge function, inwith order energy for canonicalexpansions Fronsdal in fields terms of particle modes in fields with Lorentz spins them into full solutions with black hole modes formeda as Klein operator a in back-reaction. a linear vertex in Vasiliev’s equations, which hasspin the effect gravity of [ converting the star-product algebra: evidently, theclose twisted under projectors star are products notPhysically, onto this idempotent, the implies but projectors, rather that, differentlyonly which from in solve black their the hole turn linearizedscalar modes, close particle equations, the onto modes scalar and themselves. are particle not injected modes into the the full Weyl zero-form, equations the as nonlinear well. corrections dress Therefore, once in the Weyl zero-formterms amounts of to rotationally-invariant rank- anP expansion of its twistoranti-supersingleton fiber weight spaces for space positive dependence andpressed negative in the spin degrees of freedom. These elements form an indecomposable subalgebra of matical methods that attreatments first of sight gravity, may but appear thatthe to are, unfolded be formulation. on distant As the frominto mentioned other those a above, hand, used gauge we function, in natural thus and ordinary from absorband construct the the a twistor spacetime deformed point space oscillator dependence of connection. algebra view from the of Weyl zero-form them as higher spin black holes in this1.3 paper. Summary of ourIn this new work, results wein shall [ extend the spherically-symmetric, static black hole solutions found various non-trivial ones consisting ofparticular, compatible as combinations found of in regular [ specific presentations. regular In presentation for the projectors, which ensures associativity and traceability. modes, while we leavecation. the issue Adhering of to a the proper current interpretation use of in the the latter literature, to we a shall future therefore publi- keep referring to compositions must be regularized. Thisobtained raises the by issue superimposing of how different topresentations. sectors handle field equipped Indeed, configurations a with compatibility problem separately mayof well-defined arise upon regular such dressing linear linearized combinations solutionstrivial outcome, into full namely that solutions. each field This configuration problem can may only have be either dressed only separately; the or conditions in spacetime induce classes of functions in twistor fiber space whose star-product JHEP10(2017)130 | 0 2 n ih 2 n | , where the bra is as we shall see in | 0 8 2 n ih− 2 n | , as they blow up in the interior, ]. 8 ]. These operators are then mapped via the horizon states 87 – 8 – ]. Thus, without any further fine-tuning of the 3 — since at the boundary all Weyl tensors fall off – 1 vector model. boundary states N ], to the aforementioned twistor-space realizations of operators 20 9 . We defer to a future study the interesting problem of whether one can, order 0) in supersingleton state space [ 5.2.2 > 0 We would also like to remark that while the internal connection does not belong to At first order, though, the black hole modes can be switched off, and it is possible The above results lend themselves to a holographic interpretation as follows: let us Our choice of nomenclature reflects the fact thatConfigurations this that gauge are was singular used on the by twistor Vasiliev base in manifold his may original instead works have a role to play in a more 8 9 n, n general physical context, for example in describing nonlinear density matrices. prospect of using thisspace initial requirement data. as a superselection mechanism foron admissible the twistor classical perturbation theory of his equations [ it belong to anparticle associative and subalgebra of black the holeon star the state. product deformed algebra oscillators Moreover, foras is a necessary the general for to effect combined the make ofsolution existence them the to of real yet Vasiliev’s vacuum analytic another gauge, AdSfact on field-dependent which that gauge gauge the we the function, function construct twistor solution bringing space at base exists the the manifold, in linearized the Vasiliev level. gauge as We being consider nontrivial. the This raises the gauge fields, thereby obtainingand defining field a configurations quasi-local Fronsdal expandabledual branch of over of moduli only the space, free particle that large- can modes be a proper candidate any differential graded associative algebra, the deformed oscillator variables formed from distinct Lorentz spins. to make contact with thesection usual perturbative scheme inby order, Vasiliev gauge, remove the Type D modes from the twistor-space connection and the spacetime forth higher spin black holecan modes, that be blow thought up at ofand the the as origin. theory In linearizes, this sothe sense, that the twisted one projectors projectors can can think be ofwhere preparing thought the single of fields Fronsdal as coalesce fields — into while collective degrees of freedom that can no longer be assigned start by injecting into thewith bulk conserved higher the spin states currents, conjecturedresponding by to to expanding be massless the dual particle Weylinto to zero-form modes the the over of bulk, operators free arbitrary the cor- vector (integer) Weyl model tensors spin. become Following strongly these coupled, modes and the non-local vertices bring supersingleton-supersingleton and supersingleton-anti-supersingleton operators,corresponding the to latter black-holeinitial modes data, [ thesecond backreaction order from of the particles classicalthe perturbation produces innate theory, which spacetime black one non-locality may hole of viewfill the modes as in Vasiliev the a system. already details manifestation of of In at the the above the bulk sketchy derivation of of the this paper, interesting backreaction we mechanism. shall Schematically, the massless particle modesFronsdal are theorem first [ mapped, in accordance( with the Flato- aforementioned linear vertex to an outgoingan operator anti-supersingleton of state. the At form the second order, the interactions produce an admixture of particle modes into black holes modes by means of a Fourier transformation in twistor space. JHEP10(2017)130 ) C and , and B , C . The paper provides the 6 3 regular presentation , thought of as a classical , we apply the Ansatz to C 4 describes the method used for ; useful properties of the Klein at the linearized level; the latter , and then a compensating field C A ). We would like to stress that, while . In section 2.68 )–( – 9 – Vasiliev gauge 2.64 is carried by a differential associative algebra Ω( C ; finally, appendix B contains the analysis of the spacetime behaviour of the vacuum gauge function 5 holomorphic gauge ]). Our conclusions and an outlook are in section 84 , 13 , 12 , , we spell out the properties of Vasiliev’s equations in four dimensions that 9 , 2 8 The noncommutative structure of , respectively. The fibers are symplectic, and the higher spin algebra arises naturally structures, though the requirementwithin that this higher category spin ofand quantum gravity constructing geometries admits new may a models. prove global fruitful formulation in constraining theconsisting theory of symbols, which is a class of differential forms on ometrically, these are horizontalto differential as forms the on correspondence ative. bundle space, We space, whose shall that fibers denoteY we as the shall correspondence well refer as space,as its base symplectomorphisms, base manifold while and are the its noncommuta- (equipped fiber base with can space an by in integration general measure). be a Further differential below, Poisson we manifold shall largely trivialize these Sitter spacetime in the presence of particle and2.1 black-hole modes. Correspondence space Vasiliev’s higher spin gravity is formulated in terms of a finite set of master fields. Ge- of Vasiliev’s equations isour given solutions in only eqs. rely ( some on global the issues that locally maywell defined be as Vasiliev useful in system, in extracting providing wea Fronsdal them shall fields related with here from additional a also the physical setpropose interpretation, full address of as system. will boundary lead conditions, To this to including end, a ordering we well-defined prescriptions, shall perturbative that introduce we expansion around asymptotically anti-de 2 Vasiliev’s four-dimensional bosonic models In this section, we describegauge the fields basic of all algebraic integer structure spins of in Vasiliev’s four equations spacetime for dimensions; in bosonic particular, the original form example [ contains three appendices: ourfields conventions in for anti-de spinors Sitteroperator and spacetime are the are contained gravitational given in background inconstructing appendix the appendix deformed oscillators. algebra in twistor space.Weyl Section zero-form and thegauge internal by connection first in switchingdependent the gauge on gauge function the reached leading fromgauge to permits the the a holomorphic perturbative description in terms of self-interacting Fronsdal fields (see for In section are relevant for constructingsolution and interpreting Ansatz our in exact athe solutions. specific aforementioned Section generalized projectordemonstrate the algebra backreaction in mechanism a and the specific associativity of the deformed oscillator 1.4 Plan of the paper JHEP10(2017)130 ): · ( (2.1) (2.2) ? ) · , , as Vasiliev’s ) are sections, C  b g b d finite  ). b f ? Y ) ( ]. For the application to b (thought of as a compact f 2 C 77 L – deg( ]. 11 74 1) , which are determined on-shell − 106 B – + ( ) then b g 104 C ? Ω( ).  C b f ∈ b d h b  b g, = – 10 – ), so does the star product; we thank Giuseppe de Nittis b f, 2  R b g ( , which one may think of as the infinite-dimensional 1 L Y b f ?  h . b The noncommutative structure thus amounts to a dif- ) are used to distinguish them from elements on various subbundles b d ? C ) b g , b f ? ) and a compatible associative binary composition rule ( C = 0 in order to yield finite star products and traces; thus they should ) is dictated by the boundary conditions of the theory. We shall ) = ( b C f Ω( ) and have curvatures in Ω( h b ), that are (finite) deformations of the de Rham differential and the B 2 in the Weyl zero-form must be elements in C As for connection components, it is instead assumed that they give rise C b d → b g ? Y Ω( 10 ( ) indicates that the it is in general a nontrivial deformation of the de Rham differential, star-multiplied with component forms on C ). b d → B b Y f ? ( ) 1 : Ω( C L b ], i.e. the elements must be real-analytic in order for the Lorentz tensorial com- d , the integral version of the star product is equivalent to the twisting of the convolution product 2 Ω( 87 R [ ⊗ 4 As for the space of functions on The choice of Ω( ) The hat on On C 11 10 by a cocycle given byconvolution a product phase closes factor, for whichfor functions does pointing in not out affect this the fact convergence to of us. the integration;whereas thus, the as hats the on thethat elements will in Ω( be introduced below. wedge product, respectively, such that if terpretation; assuming that thar thesethe inner functions products on are induced via theNoncommutative trace operation, structure. ferential Ω( AdS ponent fields to be well-defined.degrees of In freedom the of case the of theory,a we the shall positive Weyl furthermore zero-form, assume definite that which sequi- its containare function the or space diagonalized, local admits bilinear in order form to in attempt bases to where provide compact the theory higher will spin a generators quantum-mechanical in- to well-defined open Wilson lines,using which deformed can be oscillator implemented algebras; in for noncommutative example, geometry see [ representation matrices of thespin theory, we algebra shall that consider lend modules themselves of to the the underlying unfolded higher framework for harmonic expansion on consider solution spaces in whichfunctions the on master fields canby be differential expanded constraints. in termsfinite As of integrals for a over the basis section of belong components, to it is assumed that they have order to construct closed andbuild central cocycles elements gluing in sections positive tosections connections. form and degree We connections shall that use alike, can thewhich though, be term means used more master that to precisely, field the the to zero-formsact refer elements must faithfully to in be on Ω( bounded, Ω( while the connections are assumed to may obtain these structuresclassical by integration measure deforming for the smoothspace) and wedge bounded along product, forms a on de semi-classicalhigher Rham differential spin Poisson differential structure gravity, and however, [ ittheory makes is explicit crucial usage that of non-trivial these roots deformations of are the unity, that have no classical limit, in manifold, equipped with a star product, differential and trace operation. Formally, one JHEP10(2017)130 (2.5) (2.6) (2.3) (2.4) is not . The Y b f , = , playing the † ) Y † ) b f (( ], and assume a the quantum atlas by charts and corre- 75 in order to project the star , C , Y 74  †
b f ] along  78 ; conversely, we assume that the b d S = † , ,  b f ,i.e. the kernel of inner derivation along vertical , b d C to be symbols of (super)traceable operators ) compatible with the quantum geometry, requires a  C Y X × Z – 11 – As for the bundle structure of the correspondence S ⊂ B ], which combine bundle isomorphims (including ∼ = → C −→ B , , † Kontsevich gauge transformations can be used to B 62 to be flat and torsion free [
Y b C f used to expand the sections will in effect be assumed to are not removable by going to a “finer” quantum atlas. As has already been remarked above, although Y ? S to be a symplectic manifold with global canonical coor- large † to be closed (hence compact) while allowing the (higher 12 4
b g R B ) b g ∼ = Y )deg( b to be trivial with f C deg( ). We shall assume that there exists a cover of 1) C − Ω( viz. = ( ∈ , with a compact topology (at the level of the trace operation); further below, we † † b g  Y b g b f, viz. b In order to describe asymptotically anti-de Sitter higher spin geometries, we shall take We furthermore assume f ? The elevation of the space of horizontal forms on  12 vector fields, to a differentialdifferential graded Poisson subalgebra structure of with Ω( specialproduct (abelian) and Killing a vectors special [ vertical top-form to project the quantum differential. such as, for example, conformalfor classes conserved of currents higher on spin gauge the fields dual corresponding conformal to field sources theorythe side. differential Poisson structure on bundle structure of modulo gauge andatlas ordering such artifacts, that that thesingularities in is, connection the is we connection on smooth assumeWe away remark that from that, there ingenerate exists physical this a singularities sense, quantum in the connection whose structure define boundary states, in Fock spaces (defined using Weyl order). spin) connection to blow up on a submanifold we take the fiberdinates; space the structure grouprole is of higher thus spin the gaugecompact, group group. the of class of symplectomorphisms functions on of provide shall achieve this by taking the functions on formations. Topology of correspondence space. space, corresponding similarity transformations (includingby transitions Kontsevich between gauge charts) transformations are [ higher given spin gauge transformations)ric) with polyvector re-orderings fields. ofobservables The symbols given resulting generated by ambiguities functionals by are that (symmet- are factored invariant out under at (small) the Kontsevich gauge level trans- of classical operation for all sponding bases for local symbol calculus, which one may refer to as a These operations are in addition assumed to be compatible with an hermitian conjugation JHEP10(2017)130 ∼ = 0 and (2.7) (2.8) (2.9) Z S (2.10) (2.12) (2.11) . p N p ) valued in Z ( 1 combined with L Z . Furthermore, we ] for the definition. ], as we shall describe that are real-analytic W 107 . 2 107 S × , ; see [ 2 N Y S that are forms in × , we shall keep the time-direction × P . We propose that requiring the , 4 X normal order N T , ) R p 8 T In providing the (perturbatively) exact ∼ = R × 3 2 Ω( 1 S S ∼ = S ⊗ 0 × × ) ∪ Weyl order 1 2 Y × Z X S S S ) is given by integration over – 12 – T := Y × Z ∼ = ∼ = by inner Klein operators [ × P T Z X Y S := ) = Ω( p 0 C T × Ω( 1 , S = S X × T given by a regularized integral over = C W to consist of asymptotically defined Fronsdal fields, defined order by , at generic spacetime points. As for , that is, they can be expanded in the basis of monomials in the canonical 0 0 by allowing the connection to blow up at its south and north poles, ) consisting of symbols defined using 0 S of the Weyl algebra on 3 T T p T S W are planes passing through the origin of ) consists of symbols defined using N , is thus obtained by adding points at infinity to T S ]. In other words, we have P T As for the status of the boundary conditions on the connection, we shall implement 107 [ , respectively, corresponding to the boundary and origin of the anti-de Sitter spacetime 4 N quasi-local Fronsdal branch arises upon further requiring smooth connectionsthem at only to thehigher linearized orders, order. and, It ifirreducible remains gauge so, to equivalence be class) whether in seen they view whether are of they the non-trivial can ambiguities (in residing be in the reached the at sense form of that large they select an where connection at order in perturbation theory, yieldsthe an implementation irreducible of gauge thistransformation, equivalence as class boundary will of condition be connections; requires discussed further the below. aforementioned We large shall gauge furthermore propose that the decompactify p background, i.e. we take a trace operation on We shall then apply athe large algebra gauge Ω( transformation, andat convert the the origin sections of tocoordinates elements in on periodic by working with a compact spectrum of states realized in where Ω( an extension further below. The trace operation on Ω( equipped with global canonical coordinates. Ordering schemes and boundary conditions. solutions, we shall start from an Ansatz in which the sections belong to the algebra where is commutative, and is a non-commutative spaceR obtained by adding (commuting) points at infinity to where JHEP10(2017)130 (2.14) (2.13) valued that is a X p , ); for a brief b ), at least at 0 A| ) ˙ Y α T ( ¯ z 1 − L 13 , Large gauge trans- α 2) z , ; ˙ α , ¯ y ˙ = 1 α , z ˙ α α y = ¯ α, † ) ); ) (by going from normal back to α ˙ ) = ( α z T , the restriction of the connection α ( ) with the standard trace given by 0 α, Z ; T α ∈ X = ( , Y p ( , or more precisely, projecting onto the deformed oscillator algebra α ˙ α y X . = ¯ A to , † ) , defines a higher spin connection on ) using ( – 13 – with a B α α Z 0 y C Z ( T ; µ x from , C = ( µ x M = † ) . µ x We coordinatize ( ,X 5.3 in terms of bounded component fields, which may seem surprising ) α ) can actually be mapped back to Ω( , which would require the symbols to be elements of ], is that the relevant Type D sector, which is infinite-dimensional in 0 ) and that in its turn is assumed to admit a Weyl ordered description 0 0 . However, as we shall see, these can be summed up and converted to 1 W Y is assumed to equip T ) that are constant on T ; N T 0 0 p M T X } × Z = ( p { ), namely, the Vasiliev gauge condition to be spelled out below, ensures that the 0 ). M Ξ to T T We would like to stress the fact that Vasiliev’s procedure is designed to describe inter- Our spinor conventions are collected in appendix C -transformation to the ordinary massless particle spectrum consisting of lowest-weight 13 2 with reality properties Z spaces, as we shall recall below. Local coordinates. than to obey any quasi-localityof conditions singularities in in spacetime. spacetime Whether toable the distributions model resulting in for conversion twistor space blacknoted provides holes already a remains in physically accept- [ tohigher be spin gravity spelled already in out the in metric-like formulation, more is detail. mapped by A means of promising a fact, simple Lorentzian spacetime. Dually, foron a generic point subalgebra of Ω( in Ω( actions in spacetime that are dual to deformed oscillator algebras in twistor space rather restriction of the connectionelements in on Ω( in a higher spinthat Lie the gauging algebra, of todal the be fields higher defined with spin particular algebra, below. self-interaction or upon unfolding The taking procedure, Vasiliev the yields gauge spin-two a condition subsector set to implies of describe Frons- a Higher spin gaugeformations fields are and required deformed inconditions oscillator in order normal algebras. to and construct Weylin order. connections Ω( In that normal obey order, the the corresponding dual gauge boundary condition regular elements in the extendedthe Weyl linearized algebra order. (given Another by problemquasi-local delta that Fronsdal we functions branch shall on can not be address in equivalentlyscheme. described any To detail using this is an whether end, entirely the normal-ordered itthe would integral be over natural todiscussion, equip see Ω( section the sections in Ω( Weyl order). We remark thatover this the would basis imply the of in Weyl zero-form the is case actually of expandable singularities the at black-hole-like solutions, whose separate linearized Weyl tensors have Kontsevich gauge transformations. We shall also leave for future work the issue of whether JHEP10(2017)130 , ) v (2.20) (2.21) (2.22) (2.15) (2.16) (2.26) (2.23) (2.24) (2.25) (2.18) )) with +¯ , ¯ z } ˙ denote the y, z v, α ( v b − f . . ( z = ¯ ; )) M , , , v † ∂ ) , . Normal ˙ +¯ α M αβ y, z αβ − β − β O ¯ y v ( Normal = 0 a a (¯ iε iε 2 v, ? O b + α − α dX f ] − − a a ( + β , = β β y α = = ( , z z z v 2 α α α =0 b − β − β f y z y α [ = Normal ) ¯ † ? a ? a u = = O dY ) | β β − b + α − α α )) d ¯ v z , ( ? z ? z = ; (2.17) loc u, y, z , α α b ( αβ f + , , ) 1 i b hor ˙ z ∧ α f α | ) to the operator 2 ; ( z b u d u − M = 0  = ¯ +¯ y, z Ξ ? ( = b ¯ α d i y † b f y ) ? b Normal f , ] ( ˙ u, α = , β ) O . To be more precise, letting u 1 2 – 14 – b + f (¯ M Y , z y ; ? , a Ξ ( α := b d 1 z , b )) = [ h M f αβ , y  α , z α Ξ a ) iε u b , we shall use Vasiliev’s normal ordered star product, ˙ d αβ αβ α y, z 0 X, dX ¯ ( u + , a ( = iε iε ˙ T α , 2 b f b v † β + β + f ) + + +¯ a a normal order αβ ? α = α β β α + α − On ) (2.19) ) u u i y y ¯ a a z α α α =0 v y z α ( = = y, z i = 2 ( e ¯ ):( y,z, = = β + β + dY ? 1 ˙ | ¯ ] v α b f β β b 2 y, ¯ v β f ( ( ? a ? a , 2 , y vd α ? y ? y b = 4 α + α − loc f 2 v α ) a a α α ; ? y π ˙ z [ y ) ¯ α ud hor Normal ¯ z 2 ¯ | u ) defined in normal order, we have (2 b , O f α ud 2 y, z u ¯ y,z, ( ( d b { f y, R ( R the anti-holomorphic variables. Equivalently, in terms of the creation (+) and anni- their complex conjugates. We also have 1 = b Z f R = R Thus, the star-product formulasymbols provides defined a in realization the of above Wigner the operator map product that in sendssymbol terms a of classical function one has idem hilation (-) operators that is, all auxiliary variables are integrated over the real line. In particular, it follows that where the integration domain respectively, by Star product formula. given by the following twisted convolution formula: where the bracket is graded, and we choose a symbol calculus such that in what follows welocal shall suppress representatives the of wedge horizontal product when forms ambiguities and cannot the arise. differential The acting on them are given, idem and canonical commutation rules JHEP10(2017)130 ) C (2.33) (2.31) (2.32) (2.27) (2.28) (2.29) (2.30) ). We z ( λ is an inter- f , ) y F . )) ( z λ ( )) f z 2 ( λ f f ] and ) ( P , , y 9 ( , ). The resulting fields )) )) 1 ¯ b J 7 y z f , ( ( ( 4 b ], and, more generally, Normal J, 2 2 9 f f ] leads to a twisted adjoint O – , ( ( Weyl 7 ) b 107 J O Weyl Weyl − )) = . z O O ¯ ( 0 b J, )) = )) )) f z − y z ( ≡ ( ( ( b ? b 2 A, 1 1 ] A, f f f b Weyl ( ( ( F − , b O A? b A, b Φ) Weyl Weyl Weyl + ( ) of an operator defined using normal order can O + [ b O O π A – 15 – , b ) (over some classes of functions or distributions )) d b F y, z z y b d = ( ( )) ( ( b The twisted convolution formula extends the Moyal λ y := )) = )) = f † 1 with curvature ( ) z y f := b ¯ ( ( f ( F b J b ? f ( A 2 2 b F ) ]. Consistent truncation [ b oscillators are mutually commuting has been used in the J, y b D ? f ? f ( Weyl b λ Z ) ) 107 A, O f z , y Normal ( ( λ b Φ O 1 1 ( is generated by outer Klein operators [ and f f P )) = ( ( z K Y ( )) = 2 ) = y is a one-form Weyl Weyl ( ] master fields, that can be introduced geometrically by replacing Ω( O O f C ? f y, z ( ) ( b y 107 , where f ( denote the corresponding map defined using Weyl order, we have Weyl 1 f O ), then its symbol defined using Weyl order is given by ( Z b Φ and a pair of closed and twisted-central two-forms ( Weyl × K × F O ) Normal and C O Y by Ω( nal graded Frobenius algebra [ zero-form obey the reality conditions identity Higher spinbi-fundamental gravity [ also makes use of twisted-adjoint [ The connection on transforming in the adjoint representation of the structure group, and obeying the Bianchi to group elementsinvolve and indefinite projectors, diagonalizable this bilinearshows may that forms; result these resorting integrals ineigenvalues to must of auxiliary be the the Gaussian performed bilinear original by integrals forms. Moyal-product means that of analytical2.2 continuation in the Master fields the normal order. Regularization of star products. product from the spacefunctions, including of delta function real distributions (including analytic their derivatives). functions When to applied the space of Fourier transformable where the fact that the last step. In other words, ifbe the factorized symbol as on would like to stress, however, that, in what follows, all symbols will always be given using and that It follows that Letting JHEP10(2017)130 (2.46) (2.44) (2.45) (2.41) (2.42) (2.34) (2.35) (2.36) (2.37) (2.38) (2.39) (2.40) , and ) z b d ( , . 2 π ) ) b b g g ( ( πδ = ¯ ) ; (2.43) . ¯ π b ¯ π f ) , ? π ? ( b C d := 2 ) ) ) , b b α f f z b ? τ Ω( d z ( ( ) α π π π ∈ b g ( iy , b g = τ b g ) = ) = ¯ and b , b f, f , κ b b g g ) b b d d π = 1 ) ¯ 1) b J τ y y b b ( − = exp( f ? f ? . − ( ( 2 = ) , , z ? κ ¯ b π ¯ b J, ) ) admits the following non-trivial J , y b πδ f , − b ? κ ) = ( π d τ b ( J, ) = 0 b g . y b b 2.39 b Φ] A A, π κ π := 2 ] , ,, π b b − b f f ? ) ) y b b b g ? π ). A, Φ] Φ , ) = ( ( ˙ ˙ ) , κ := α α ¯ b b b g J ) ¯ ¯ J, z z b Eq. ( b b κ [ Φ) F, y , [ ( − b 2.40 α J, b ) = ( − , π Φ + [ ( z )deg( ( ≡ b α b – 16 – b A d f , κ π f , τ − z , b z Φ , ; ; ) b 2 Φ ˙ ˙ κ = α α α ) = ( b κ , deg( ( b ) = ¯ ¯ ¯ ¯ y y D b y J ¯ π ¯ b αβ b ] , Φ := 1) J = 0 αβ iZ − ε π α ε b b , b J, − D β − b ? κ y ) implies ( J β J, ( 79 α ; b , d ( ) b − y dz α A, 3 y dz π − ; ; , ( ∧ µ µ 2.42 b g ∧ iY b Φ α x x ; ( α ? f µ τ b dz f ? y x i dz 4 κ i 4 ) = ( ) = ( − ˙ ˙ := α α − ¯ ¯ z z π ) = ( ] , , := = α α α b g z z z b Z J ; ; ; b f, ˙ ˙ α α α [ are the involutive automorphisms defined by ). The bosonic models require the integer-spin projection ¯ ¯ y y it follows that ( Y , , C π ; α α ¯ π , j µ y y Ω( π y ]: x ; ; is an inner Klein operator obeying ( 3 µ µ and ¯ ∈ is the graded anti-automorphism defined by = τ y . Thus, one may write [ ?κ x x b z ( ( f 2 z κ π τ j κ ¯ τ π π = b J idem Twisted-central and closedsolution elements. [ where where from In the minimal models, theprojection odd-spin Fronsdal fields are removed by the stronger even-spin for all which together with theeach reality rank conditions occurring lead once. to The real twisted-central elements Fronsdal obey fields the with stronger integer conditions rank, obeys the Bianchi identity The conditions on the two-form making it closed and twisted-central read The twisted adjoint covariant derivative In order to define the twisted-adjoint representation, we introduce the graded bracket where JHEP10(2017)130 (2.49) (2.50) (2.51) (2.52) (2.53) (2.54) (2.55) (2.56) (2.57) (2.58) (2.59) (2.47) (2.48) given by P ] , . 88 ) ) , , y b 9 A ) b d , , − b A P b Φ , z, b − Φ = − ( , b . f , ? b dP b κ ] ) = ( ) = (  b b b A A = = 1 , = 0 b Φ, one has [ , b , and the two-form is treated , , A, ¯ b b κ κ b b b Φ b b J A Φ Φ  , ]: ( ( ? ¯ ? b κ . ? + [ ) ˙ ) , β b Φ) b κ ? 29 α  b ˙ ( α ?n b z d b Φ) ε  ?n ( ˙ iθ y, z − β (  ,  ¯ . z := F b b Φ) ˙ f α ) d ? (  b b + ¯ Φ) b z J , ( . ∧ b b D ¯ − J b J ˙ ? π ¯ α ; b J, ? ¯ ? π = z considerations on-shell. In the off-shell formulation α b = 0 Φ = exp d ,P b b  Φ ,P i 4 A , y b Φ)  B b ˙ 2 π  +1 ( α ) = ( − n b ¯ y – 17 – Φ) b J F 2 ( ; ¯ = 0 = b = Φ) µ a priori b b ( + df , J, θ θ +1 x † = 0 idem ( , n ) n b , 2 , δ )) b F 2 b J P b J Φ θ )  f b π ( δ ? 14 ) = ( y, z represent degrees of freedom entering via a dynamical two-form. − b =0 Φ] ∞ =0 ˙ ( ∞ z, y α X B n X n b ( ¯ z F , f , b = b [ ( f , = = = b α b κ π J − Vasiliev’s equations of motion are given by θ z are complex-valued zero-form charges, which are functionals of F F ; = ˙ ) defined by α b ) = Φ = 0 ¯ +1 y b C Φ = b n ,  b D ? κ 2 δ α y, z ) f y ( ; b cannot be fixed by any f µ y, z x F ( ( b b κ ? f P Type A model (scalar) : ], the parameters in , where † b κ ? ) 107 are real-valued zero-form charges. Introducing the parity operation F breaks parity except in the following two cases [ n 2 B Type B model (pseudo-scalar) : θ = ( F The functionals 14 b where the parameters obey the sameas kinematic a conditions background as in the sense that proposed in [ The equations of motionCartan gauge are transformations: Cartan integrable, and hence admit the following on-shell and by a linear action on the expansion coefficients of the masterHence fields, it follows that where the automorphism of Ω( on-shell. Factoring out perturbatively defined redefinitions of where the interaction ambiguities. and Φ obeying 2.3 Equations ofMaster motion field equations. By hermitian conjugation one obtains where thus JHEP10(2017)130 (2.72) (2.69) (2.70) (2.71) (2.60) (2.61) (2.65) (2.66) (2.67) (2.68) (2.62) (2.63) (2.64) . ? ] . , )  b ) ˙ α , b ¯ b U, V , , , i ,  ) ˙ ) ) ˙ Z,Y β ¯ β + [ S ˙ ; o ¯ b b κ κ ˙ ( α α )  b + 2 x ? ? d ( and a covariantly constant m ˙ c M α c M ˙ α α ˙ b b ¯ z Φ Φ β b b = V S X ˙ + α ? ? ˙ − α ˙ ? b , β ω U ˙ B (0) α α  b dz ) by means of the field redefini- b + ¯ V ˙ ··· β − B − i c M ˙ α 2 ? αβ ) + ¯ ω (1 (1 = 1 ˙ , − β c α M ˙ ( ˙ αβ α , δ β α In order to exhibit more explicitly the b αβ ˙ S ? α z αβ Z,Y i i ] ω ; b 2 2 ? V, , . , , , , ω α c M x b ¯ k  V − − = ( ( ). These two objects belong to subalgebras + ? i α ) α , b α 1 = = = 0 = 0 = 0 4 [ b b b ¯ V b κ U b S V ) = 0 ) = 0 ) = 0 ? ? ? ? α , i − b ? ] ] ] ] – 18 – ˙ , U α α 2 b ˙ = ˙ b U := Φ β β α α ) b ¯ dz S b ( S = b b ¯ ¯ S b S b S b S S ( − ( (Φ A ( b αβ , , , b K α U? ¯ π ? ˙ , α α α ? π b α b S c U, M ) = ? π ) b ? b ¯ Z S b + S S  b [ b ?k [ [ Φ b b ) + Φ Φ + [ b U b κ − d Z,Y α Z,Y ) := (0) ? αβ ; b To obtain a manifestly locally Lorentz covariant formulation b ; b S Φ ˙ K, α b b Φ + x Φ + x d b c Φ , δ ¯ M ( b ( S ( ? − ? α π µ b − n U? ˙ b α V b α b U := , U b b Φ] ¯ α S b µ S α := b , b S αβ := [ b dx dZ Φ + b A d c − M c W := ( := := , the master field equations can be rewritten as µ α b b U V b b ∂ Φ = S µ  b ] δ dx 88 , = d ) that we expect to be determined by the boundary conditions, as we shall examine 84 0 , 9 T one introduces a canonicaltion Lorentz [ connection ( where generated by the internalof master Ω( fields ( in more detail in the case ofLorentz higher covariantization. spin black holes and fluctuation fields. Thus, the master field equations describedeformed a oscillator flat algebra connection on The gauge transformations now read Letting and introduce the deformed oscillators deformation of the curvaturecompose induced by the closed and twisted-central elements, we de- where Flat connection and deformed oscillators. JHEP10(2017)130 ,  ˙ ) = β ˙ ˙ (2.85) (2.80) (2.82) (2.84) (2.73) (2.74) (2.75) (2.76) (2.79) (2.81) (2.83) (2.77) (2.78) β α ˙ α ¯ ¯ ς c M , ˙ β ˙ α αβ ς ς +¯ , , , ) αβ ) ) ˙ β . ¯ b b κ κ , ¯ z c M ? ? ? ? ? .   αβ ˙ b b α Φ Φ α ς α ( ˙ ¯ b ¯ b z ? ? S  S = 0 i , , , B 1 − 4 ) ˙ ˙ ˙ γ ) γ β =0 , ˙ ˙ ˙ − − B − (0) β β (0) β ¯ Z b S , ¯ ? y = =  ? (1 (1 c M c M ? ? ˙ Y ˙ ˙ β γ γ 

α ˙ ˙ ˙ α c α αβ β ( ˙ W β c W ( ˙ b , and Φ ¯ ς b ω S ω i i c ˙ y W , γ δ , ˙ ˙ 2 2 ˙ , , , , , , , β β β + ¯ ˙ + ¯ ˙ β ¯ y (0) α − − := ω := ¯ ˙ (0) α ˙ 2 γ ∂ ) ˙ ˙ ˙ (0) ˙ β (0) β βγ α ¯ c ∂ = = 0 = 0 = 0 = 0 = α βγ M ⇒ S ˙ ˙ (0) α c ˙ M ( α ¯ y β ω ) = 0 ) = 0 ) = 0 ˙ ? ? ? c c ˙  β M M ] ] ] α ˙ ∂ ˙ α α ˙ ˙ c c W W ˙ c β α M c M β β α + ) b ω c ¯ ˙ W S b βγ S α βγ ˙ b ¯ ¯ ω ? ? b S ( β b ˙ S S ( ( β ). As a result, the master equations read ˙ ω , ω , α , ˙ ˙ + ¯ ˙ ¯ α α α π α c M ˙ ¯ + ¯  ς α  α α ˙ b ¯ ¯ ¯ S b ? π , β ω b S ? π S b ? b ¯ i S i , b S ˙ – 19 – S , , (0) α [ 1 , d αβ 1 [ [ 4 4 (0) b ) b Φ αβ ) b − r Φ αβ − Φ α β β c ς M b ˙ S α c α + + := b M − + ¯ = 0 S b ¯ ˙ S b S ˙ β ? z αβ β β ? b b Φ b αβ Φ + Φ + ˙ ¯ b b α S S ω αβ α =0 ) = ( α ˙ ω ( r ˙ ? ? β β  ( β Z c z  ˙ ˙ M α α b ˙ c α i W? α S c α W? = 1 i c 4 W? ω ω b ¯ ¯ ω S 1 − b Y S αβ 4 + ,

+ and ¯ ) r := + + ¯ + ˙ β α α  ) αβ γ c ˙ W b α α b S ¯ Φ + b i S β S ω b b β b ( 1 c W Φ + S S αβ ? y 4 ( ω ∇ ∇ ∇ d d d d ς α c 2 M ∂y ( + αγ ∂ α y oscillators, and the external part := := := ω , δ c ˙ W ∂y α α Z b Φ := := + b b c W S S ∇ ) defined by ∇ ˙ ∇ ∇ αβ (0) β αβ ) = 0 c W? ˙ and α ˙ α c M dω ¯ + M b S Y ¯ ς , , α := c W b S αβ , ∇ αβ b M Φ r ς ( b U, M ( ς δ which can be usedby to imposing embed the canonical Lorentz connection into the full master fields The fielddX redefinition implies a local shift-symmetry with parameter ( where rotating the spinor indices carried by ( rotating the are the full Lorentz generators, consisting of the internal part JHEP10(2017)130 ) b Υ 1) X × ,W > in the (2.90) (2.86) (2.89) (2.91) (2.87) (2.88) . The s ( φ T ) − s ], and the (2 -derivatives α of involving 84 Z ) = , C , φ 9 W ( – P =0 7 , Y

]. Thus, a modification 4 16 generated by Φ and ) s 83 . µ X W =0 1 y

− s provided they hold at C ˙ α s ¯ (4) or in its minimal projec- y 2 , the scalar 1 ¯ ∂ α ˙ α hs α W, ∂y e ··· s 2 X × Z 1 ; for details, see [ = =0 ˙ , α ∂ ··· µ 2 Y ¯ y b 1

A − ¯ =0 ∂ ]. α , s 1 Z 1 = 0 2 W | in the type A model, and − ˙ ∂ ∂y ˙ s α α =0 φ α c ¯ W y ¯ Y b To obtain a perturbative expansion in terms V | 2 ∂ ˙ α := ∂y ∂ α z 0) ) ) = s ), prior to Lorentz covariantization, it ensures that terms – 20 – φ ∂y + ¯ ··· , > := Φ (2 ( 1 α α s P φ α 2.87 b . V := α = Φ ˙ W ∂y α z 1 6= 2) α e − =0 s s b Υ is a Lorentz singlet with a perturbative expansion starting at the ,C Z ˙ = 0. Eliminating these terms perturbatively, using the constraint on α | 1 1 1, b Z Φ , one has − − =0 s s ¯ y P > | α µ e s b Υ, where ··· := Φ 1 = ˙ α 1 ˙ C α 1 µ ¯ b α ( V ˙ α ie z Assuming furthermore that the Vasiliev frame field + ¯ α , one proceeds by imposing the initial conditions := 2 15 b V ) ]. α X s is valued in the bosonic higher spin algebra Φ drop out from the perturbative expression for z ( µ µ ∂ , where they define a non-linear free differential algebra on 84 , φ } (4) and Φ in the corresponding twisted-adjoint representation [ W ) to 8 , hs As for the auxiliary fields, of particular interest are the pure Weyl tensors 4 = 0 Under the parity operation Concerning the Vasiliev gauge condition ( [ 2.87 16 15 Z of ( second order, leads toand non-canonical its higher twistor order space corrections derivativesΦ, to at yields the a Cartan canonical curvature Cartan of curvature for type B model. The remaining components ofthat Φ are are non-vanishing given on-shell. by all possible derivatives ofinvolving the Weyl tensors Bianchi identities; for details, see appendix D in [ that make up the generating function ( and the Fronsdal tensors ( modulo auxiliary gauge symmetries; ii) dynamical metric-like equations of motion; and iii) is invertible, the free differentialin algebra terms yields i) of algebraic a constraints remaining that set express of (Φ dynamical fields, namely remaining constraints can then{ be shown toW hold on tion Vasiliev gauge condition in normal order, afterin which a it is perturbative possible expansion to in solve terms the of constraints Φ involving while maintaining real-analyticity on of self-interacting Fronsdal fields thatvariant on is locally Lorentz covariant and diffeomorphism in- where Perturbatively defined Fronsdal fields. JHEP10(2017)130 ] 64 , 7 ]. 99 ]. Moreover, as 98 ] mapping them to the as introduced in [ 95 ) preserving the topology X lend themselves to on-shell 2.87 not ], which assumes the existence of ]. The appropriate interpretation of 61 , 60 ), which ensures finite invariants given 61 , T 60 ; ii) It also yields perturbatively well-defined X ) for generic points in ]. – 21 – 0 T 107 , ] is large in the metric-like sense [ 111 102 , then any modification of ( , – ], though it remains to adapt them to the non-polynomial W 15 94 ]; iii) Provided that the asymptotic spacetime gauge fields 109 94 ], whereby the amplitudes are to be obtained by evaluating the on-shell ], this procedure yields classical fields that do and traces over (4) gauge transformations on 1 107 B 93 , , hs ) is manifestly Lorentz covariant and conveniently removes all gauge artifacts up 15 92 ) will not affect the invariants, and hence lead to a physically equivalent description 2.87 T As for the underlying microscopic action for Vasiliev’s full equations, an alternative Secondly, Vasiliev’s procedure yields Fronsdal field interactions in the Vasiliev gauge The Vasiliev gauge is required at the linearized level in order to obtain a Lorentz covari- The following remarks are in order: connection vanishes. The latter can then be switched on and the solutions brought to the convenient gauge, we shall provideinitial a data. perturbatively defined solution for general3.1 zero-form Gauge functions The basic idea is to build families of exact solutions in gauges in which the spacetime currently under investigation [ 3 Solution method In this section, wein give twistor the space method that we based shall on use gauge to functions solve Vasiliev’s and equations. separation In of particular, variables by going to a given in [ action on classical master fieldsby (containing dressing asymptotically anti-de free Sitter Fronsdal regions) fieldsthe obtained expanded findings of over the suitable present highercomputations, paper spin may once representations. serve as an Thus, a starting appropriate point on-shell for corresponding action amplitude has been found, a problem which is classes of initial data describing black holesthe and proposal massless particles, at and the to furthermorethe level verify quasi-local of Fronsdal the theory prescription may of thus [ be as aproposal quantum formulated effective directly in theory terms [ of the master fields in correspondence space, has been amplitude computations using the prescriptiona of quasi-local [ action principlede with Sitter canonical self-adjoint spacetime. metric-likeproperties kinetic In of terms response interactions, in to anti- fieldspace have these redefinitions been subtleties, and proposed criteria gauge in [ for transformations classifying given in the twistor locality at the leading order pending further consolidation of thethat above are approach. highly non-localquasi-local in the Fronsdal sense theory that [ found the in field [ redefinition [ and further developed indefined in [ normal order areand Fronsdal that fields the order-by-order in masterby classical fields integrals perturbation can theory, be mappedof Ω( to Ω( of higher spin dynamics. In what follows, however, we shall impose the Vasiliev gauge only ant description of free Fronsdal fieldsi) on eq. the ( mass shell. Beyondto this residual order, we observe that: master fields in regular sub-classes of Ω( JHEP10(2017)130 ; (3.6) (3.7) (3.8) (3.9) (3.1) (3.2) (3.3) (3.4) (3.5) (3.11) (3.10) Y × Z b g , ? , ) 0 α n b ( S 0 α , ? b V ) ) , 1 , the general solution to ¯ κ − =0 ∞ X ? κ ? ! b g . X n 0 0 ˙ i β . We shall focus on classical ˙ b b = 0 2 Φ Φ α =0  T x subject to the Vasiliev gauge ? ? α | − ) 0 0 b S b 0 α α αβ , G B  b , S Z ) on , − B − 0 α

. b Φ =0 b i S , ≡ (1 (1 Z 2 , ) = 0 ˙ ) | 0 β ) ) b ˙ b g − αβ α G = 0 b Φ n ( L, ( ) = ( ? 0 i i 0 α α 0 α 2 2 := b b , , S 0 = S α ? π Y,Z b S b µ ( − − 0 S , 0 0 ∂ αβ =0 b ∞ =0 Φ ? b Φ 0 X n = 0 = = b Φ – 22 – 1 ? = Φ ( iC ) = 0 ) = 0 ? ? ? | ] ] ] − 0 = 2 ˙ implies that, locally on 1 0 α 0 α ˙ b ˙ ) 0 β 0 β 0 α G ) = b − b ¯ Φ 0 − S b α S b b ¯ ¯ b S b S G b g S ( µ X ( b Y S , , , ∂ ( ¯ π = , (( ˙ 0 0 α 0 0 α α α ? π b ? ? b ¯ S Φ b S S ] [ 0 0 [ [ b Φ = Z b = 1 b Φ Φ , (0) 0 β ) b + S + n =0 ( 0 , 0 0 x | b b Φ Φ -linear functional in the integration constant b Φ b g (0) b g , 0 α n ? ? b S =1 ˙ 0 ∞ [ α 0 α ? d X n b ¯ S b S 1 ) is given by = − 0 b g ) is an b Φ ) 2.65 = n for details. ( ) is a gauge function and 0 b U b Φ C , ) viz. n is a flat connection obeying ( ) and ( 0 α x, Y, Z b ( S b g (0) 0 α 2.64 b S In order to obtain solutions containing asymptotically anti-de Sitter regions with free Fronsdal (gauge)condition, fields, we use a gauge function and a vacuum condition which admit non-trivial solutions obtainedsee by appendix activating Fock space projectors on where ( and solutions that admit perturbative expansions to be solved subject to boundary conditions on ( The remaining master field equations read By a choice of coordinates, we may assume that fact that the commutative natureeqs. of ( where Vasiliev gauge by means of large gauge transformations. To this end, one makes use of the JHEP10(2017)130 W , as T , and (3.12) (3.13) and C . X ; (3.14) X × Z =0 Z ;

and the nature of ), respectively, are  Z  Z on 2.86 h.c. ). (0) 0 ˙ + β ˙ b α µ V . At the semi-classical level,  ¯ ω b ) and ( X G , ), which describe local degrees ? ) and the gauge function, which Y , αβ µ 0 0 ( α β ) belong to an associative bundle 2.91 0 ω b b S S α 3) , , b S 0 ? , ), the generating functions b Φ 0 α 0 b Φ b S to be of physical relevance it has to be SO(1 ? / c W, 1 b G − 3) and Φ , b G ] and references therein): L 1 ) is necessary at the linearized level but optional + – 23 – β for generic points in SO(2 3.10 , T ? y α =0 ; y ¯ y X → , X  : ∂ | =0 L αβ Z L

that eq. ( ω  , which affects the asymptotics of the fields on ∼  ) } ) is imposed as to determine ( b i 3.2 G 1 4 ( (as a functional of =0 Z (4); for the sake of simplicity, we shall choose − 2.85 b ? π 1 G 0 b G describes anti-de Sitter spacetime. hs b Φ X ×{ ∂ ? | ? d ) are real-analytic on b 1 α 1 exp G b − ? dL S − , b b 1 G G b Φ −   L X → : = = c W, with well-defined invariants, whose construction is beyond the scope of this work. L C of freedom; condition modulo proper gauge transformations,functions that is, the space of boundary gauge W X star-product projectors entering via the flat connections windings contained in the transition functions entering via the atlas for the degrees of freedom entering via the homogenous solutions to the Vasiliev gauge the constants entering via the parametrization of Φ In summary so far, the gauge function method gives rise to solution spaces that depend We remark that at the classical level, the fields ( Having obtained We recall from section (i) large gauge tranformation (ii) (iv) (iii) In what follows, we shall activate (i) and (iii). stronger conditions arise from demandingover that ( on the following classical moduli (see also [ finally, the condition ( must belong to anlong associative as algebra, ( may be singular or given by distributions on given by their definition do not affectremove any physically observable, irrelevant but coordinate that singularities may and nonetheless other be inconvenient gauge useful artifacts. in order to of the pure Weylfields, tensors, including the including spin-two frame the field, physical defined scalar in eqs. field, ( and of the Fronsdal gauge the physical observables of theWe theory, would one may also argue like thata to it stress can be that imposed insome to observables order all of for orders. the theory.classes The formed space by of such factoring transformations out consists the of equivalence proper (or small) gauge transformations, which by such that at higher orders, though under extra assumptions on the topology of where JHEP10(2017)130 (3.25) (3.20) (3.21) (3.24) (3.15) (3.16) (3.18) (3.22) (3.23) (3.19) . 0 )) now read b ¯ h is the sum of a = 0 3.5 † b V ) 0 h b , , ( , ¯ z z ¯ y ¯ κ . ¯ κ ? ? κ ) , , ? n (see eqs. ( ˙ , ( α ) Ψ Ψ ) 0 ˙ z β . V ? ? , noting the residual symmetry b ¯ h ˙ (¯ ), while the latter can be seen Y 0 α . ¯ = 0 y ) − ¯ n b ¯ κ F ˙ , ? α ( for the hermitian conjugate; the Ψ( Ψ) Ψ) } ) ) (3.17) ? ) = 3.20 = 0 z V 0 ? ? z z 0 ) = ≡ ( (¯ b ¯ h that can be expanded perturbatively ? ) − ) ) and ( , κ Ψ y n ˙ n n ) ( (Ψ ˙ ( α ¯ π ( ( α Ψ] Z idem = n α ( B B 0 , V αβ α ? κ ). One can show that it is in addition V † ˙ ( b β V F ) ?V ? ˙ ¯ [Ψ π Ψ := Φ )) αβ α {   z Y , ? ( 3.15 ?n ?n = 0, i i 0 4 4 ) Ψ( n h 0 b )) )) ) can be solved formally by factorizing the de- ( − − +1 α b , – 24 – , , Φ Y Y ≡ holomorphic gauge V ) 0 ?n α y 3.6 = = ( n ) ) = b ] ] 0 ( Ψ( Ψ D α 0 ˙ (Ψ( ( β 0 β Y h b ? κ )–( V ( b ( ), and we have implicitly defined V ¯ 0 b 0 =0 V ∞ =0 =0 ∞ ∞ − X n ? X X n n , 3.4 ? defined in ( † (Ψ) = Ψ 0 α = [ α 2.44 0 [ ˙ ¯ π 0 b 0 ) = V ) = Φ ) = ) = ¯ b ) π V b Φ , π n Ψ := Φ + = 0 and ( 0 α ) α + ] ¯ 0 z b ˙ V ] β α D Ψ = Ψ Y,Z Y,Z Y,Z , ˙ ( 0 β b ( ( ( α V 0 0 ˙ π Ψ b , and assuming that the internal connection α 0 α ¯ b α F V [ ( b Φ b 0 V b α Z ∂ V [ ˙ b ¯ h ∂ ? ) and are defined in ( , z ¯ y ). The remaining constraints on Y κ (Ψ 0 h b 3.21 and ¯ = 0 y b κ N and using ( former follows immediately fromby holomorphicity rewriting and ( We shall refer to the aboveunder Ansatz gauge as transformations the with holomorphically factorized gauge functions In this gauge, one has As for the internalimplies connection, that the bosonic projection combined with the holomorphicity The reality conditions imply that while the bosonic projection gives where The deformed oscillator problempendencies ( on holomorphic and an anti-holomorphic one-formin on the zero-form initialpossible to data assume Φ that the zero-form is uncorrected, leading to an Ansatz of the form 3.2 Separation of variables in twistor space and holomorphicity JHEP10(2017)130 (3.30) (3.31) (3.32) (3.33) (3.26) (3.27) (3.28) (3.29) . . ) . − β ?n z u w α Ψ + ( 1 ), as the following ? +1 − u t t z i 3.5 e w . 1 , α := 2 Z +1 − z 2 t t i 1 αβ e ≥ − α n D . )). Thus, setting aside the subtler z i , z z , )) 2 1 , 1)! ( 2 , , w f 0 − ] using the extension of the ) ) − , , n 2 ( −   /t n  1 = 1 iθ ( ( π i , e z z = z n )) 1)! − κ κ ( ˇ 2 κ = = 0 = 0 ? 1 + ) ) αβ ] ? z := α α − /t ? z ˇ + D κ z ∂ ∂ (log(1 i b n ˇ β , κ ) ? exp ), i.e. 2 ( + − b κ q , z z ) 1 ( = β β β α z − ] as to include non-trivial flat connections − in Weyl order has the same properties. ( ∂ ∂ ) z αβ z = 1 (log(1 B [ 2 β β 0 3.30 ,V dt  , α 1 2 + 1)  ) 2 α α ? f 66 b z − α = 0, and is given by a distribution that is not i p V 4 t , ( D D ˇ -powers of Ψ yields z κ ( α u ) = 1   in ( ˇ ? κ – 25 – ? ( ( n V + + 1 i i 1 ( dt , − h α , + 1) α α − ?n z z V ] t n = u α α Z (1) viz. ( β + α − ) in z ) = ) = (1 + = Z α n ,   q z z V 1 α 1 ( ( z , one has = X  α + z z −  ? z κ [ = z p b α 2 Z ) u ∂ w 3.25 :=  i  ? κ ? κ . For the sake of simplicity, we shall henceforth take 1 2 n ( − z (0) ? z ] given in [ ) 0 z a − α  )–( :=  z z e C κ b + , appearing in the right-hand side of ( ( S ˇ 64 κ b 2  z 1 ). z ) exists as a distribution, it follows that )     κ n κ ] z ( − 17 2 ( 3.24 β z / ]: n 3.29 κ V ) := 1 , w 1  α obeying + [   ( z  0 z i 2 ∂ w  α → κ i  / n  u 1 − = 0; thus, the symbol of = 2 e  ) α  1 n z =1 ( := lim ∞ α + X n z 0 V ], letting i κ 1 →  = 2 = lim 0 α z b V κ ; for details, see appendix Next, we shall choose a particular solution for the internal connection. To this end, Following [ (0) 0 17 α b Assuming that ˇ that is, it obeysexistence the issue, defining we property are of led to ( real-analytic at which are solved by the coefficient of Ψ The above particular solution obeys Indeed, expansion of eqs. ( Fixing the residual holomorphic gauge symmetries, we can construct the particular solution that allows us todelta represent function sequence [ full interaction ambiguity and non-trivial vacuum connections on we use a spin-frame to define conjugate variables non-perturbative method of [ V stressing that there is no technical obstruction to generalize the results as to include the which are formally equivalent to those solved in [ JHEP10(2017)130 )- C can that , A few (3.34) (3.35) (3.36) (2 . ?n H sl 4.5 is captured we must first  I (1) , b ) V space by the inner ∓ z Y ( . Therefore, the solu- δ ) 2 , ). It enjoys the following  , the internal connection z ∓ ( 1 θ iz oscillators is instrumental to 3.29  Z = , π∂ − z z -dependence of κ z + and = 2 z ! = 1 Y ∓ ! +1 − z t t )  space, with auxiliary integral represen- normal order i I ∓ 0 e z cannot be written on a manifestly Z z  i  2 ( ), while the second one requires more care, δ i) − ) , and we have shown explicitly how the poles e dt Z  0 1 + 1) 3.34 z t – 26 – + , ∂ ( -space non-analyticities at first order. Thus, the ). ( 0 δ Z 1 1 lim → 0  − = 1 3.35 = 0, which follows from ( Z = 0. Thus, in order to differentiate dz 0 At first order, the   − I  z ∓ z dz z ∓ z + 0  z ) covariance and role of boundary conditions. Z iz z i  0 C

− are distributions in Z , e ) := 2 for generic spacetime points once the vacuum gauge function  z

+ (2 ( as deformation term in the equations, which results, as explained 0 ii) Z  π∂  sl z → ∂ I  κ in = 2 =  ) is crucial in order to evaluate star-products involving the deformed ). The price one pays is the explicit appearance of the singular Dirac I dependence in a universal way, irrespectively of the precise form of the Y  ( 0 Z ∂ 3.33 ), taking values in the extension of the Weyl algebra in and its hermitian conjugate, as anticipated in section y κ real-analytic is not differentiable at 3.30 -integral reflect themselves into t ∓ As we shall see next, choosing the initial data to correspond to massless particles As a consequence, the deformed oscillators /z space using standard distribution techniques. is switched on. This facilitates the construction of a gauge function correction be rigorously defined.becomes Moreover, uponL going to brings the solution space toFronsdal fields the arise Vasiliev gauge, in at the least asymptotic at region, the as linearized we shall level, where examine thus further below. tion in Weyl ordering isin not real-analytic the in integral form ( oscillators, and it indeed yields an associative algebra; forand the black proof, holes, see section it follows that Ψ belongs to an associative algebra in which Ψ Z covariant form; and they tations ( Kleinian with, we note that thesolving factorized for dependence on the the initial datum Φ delta function element above, in the necessary introduction of a spin-frame in order to integrate the equations in which yields the second equation in ( Breaking of manifest observations on the form of the solution space described above are now in order. To begin properties of which the first oneas clearly 1 follows from ( rewrite it as a differentiable distribution, for which we use by the distribution where we have used lim A twistor-space distribution. JHEP10(2017)130 , ), P r P (4.3) (4.4) (4.1) (4.2) ( P ], corre- − 1 = (4) algebra, (2; (0)), to the 0 1 3) [ D . , hs iπJ ) form separate (2 − ? 1)) so 4.1 ] on squares of the s, ? e r 87 [ and the parity map , viz. 0 ?P y + 1; ( 18 J 0 to corresponding highest- s + ( ? κ , π E iπJ ? ) , D e (2; (0)). In particular, the reflector )) 1) )  ( ) s D ,... − 2 − , D ( M H =1 (1; (0)) and parity odd in ) to belong to a reducible twisted- ( s := + 1; ( D π Y ) s ( ⊕ 0  ( ⊕ ( )), respectively, ) D S ) refers to the eigenvalue of the involution 1 2 . − ( ,...  P ) into two submodules; for further details, 2 H , 1 (2; (0)) (1; ( , ⊕ M 4.1 , D D ) ) =0 – 27 – s  ∼ = ( (+) ∼ = 2 S H ⊗ 2 ( ]. As far as parity is concerned, the reflector preserves the parity ⊕ ⊗ π ) 87 ], we shall take Φ  ⊕   ; (0)) and ( where the gauge function is equal to one; the inclusion of 2 1 1 2 D 87 ( , (+)  ; (0) X 1 D H 1 2 := 1; )    ( D D H   . . ∼ = ∼ = (4)-orbits. The superscript ( refl refl 1 ) − hs (1; (0)), while it reverses the parity of the states in (+) ( consists of operators with distinct eigenvalues under left and right star- D 0 . Thus, this involution commutes with the action of the full D D r 0 consist of operators obtained by acting with reflectors ]. In particular, we shall use projectors on supersingleton states that correspond to rotationally is a spatial angular momentum chosen as to make M ) 0 20  J ]. ( on D 87 The reflector sends elements in the direct product of two supersingletons to operators acting on the 18 is therefore reflected toproduct the of two left supersingletons, action that decomposes oftheorem into the [ a sum higher of spin masslessinvariant particle algebra massless states on scalar via the particle the Flato-Fronsdal states correspondingof [ state the in states the in tensor maps the rotationally invariant states,aforementioned which projectors, are parity all even of in which are even under idem and as consequence itsee [ splits the module ( supersingleton Hilbert space. The twisted-adjoint action of the higher spin algebra on such an operator These are lowest-weight spaces withweight spaces positive with energies negative sent energies. by twisted-adjoint The resulting eight spacesgiven in by eq. the ( composition ofwhere the twisted-adjoint action of ( where and scalar and spinor singletons 4.1 Particle andExtending black hole the modes approach ofadjoint [ representation space consisting of eight unitarizable irreps as follows: modes already at secondexact order solutions in in perturbation twistorover space theory. a can special We be would base thoughtnontrivial like point of gauge to in as functions stress solution will that be to the the the topic full of set the of next equations section. In this section, wecase examine when the exact Φ twistor-spacemultiplication solution by in the holomorphic generatorssponding gauge of to in the linearized the massless compact scalarUnlike Cartan in modes subalgebra a and local of spherically field symmetric theory, black black hole hole modes modes. arise as a back-reaction to the scalar particle 4 Internal solution space with scalar field and black hole modes JHEP10(2017)130 y κ (4.9) (4.5) (4.6) (4.7) (4.8) ), the (4.10) . ,..., ,..., 4.21 2 2 3)-action ,   viz. , as well as an , , higher spin , ]. In other 5.2.1 | (2 1 1 E 87  so   = 0 1 2 = ? =  )] . 1; ]. For simplicity, we ) E ( E , n , n n h . 87 ( i | | , n P n i 1 -multiplications with e , i P  supersingleton projectors  P ? n | 1 rs 1  , n ), depending on n 1) | 1 2 | M 2 − = 2 [ and [ | −  | − E,J ? ]. Indeed, via the linearization 1 symmetrized doublet indices. n n ( ; (0) ), and of the third of eqs. ( | | } 1; 2 ) 1 2 ) = ( 20 4.2 ) realizes the (left) n  2 is the doublet index of the spin-  twisted projectors spherically symmetric 1 n | − E ; , ; , 2.55 E n e |  ( , P massless scalar field [ n ( 2 2 ( n P n | n ih  ) behave as enveloping algebra real- n h ∼ = 4 P P = 1 2 n P E , 1) h− ) i ( P ; (0) static 1) n E, |− and vanishing spatial angular momentum E − ) to the rank- AdS ) 1 2 n { 4 P | |− n ( E  n i ( | ∓ denotes  ( ), 1) i |  D ) = ? 1 ], as will be shown below in section ( 1) |− ) )(1 – 28 – E 4.9 ∼ = π |− 3) on the n ( | |− ), these are required for the non-linear completion n , ) were referred to as symmetry-enhanced, in order to dis- E ) ( E 19 | ) 100 ⊕ n i ( | 4 E ) i ( E  (2 ? π ( i n (  4 ∓ i  n ( ) viz. P so . . . i S 1 P ∓  ( E D 2 2, i ( 1 , i.e. π 1 / 2 n n | − i 1 ⊕ 8 exp( 2 n | − ) | over ), this projector provides the initial data in twistor space for ∓ n  − P 0 | | − (  ) = 0 = ) ; 1) = n 3.1  S | E 2 E ) = ) = 4 exp( n ; ( ( | 2 n n E E n | − (3); for further details, see section , given by the sum of the projectors onto the (anti-)singleton states | ( ( ∼ = P n P | so ], the projectors ) ( ∼ = ? 1 i 2;(0) 1;(0) ,... are alternatively odd or even under E y  . ( E? 2   n J 2 n e  ? κ P , 2 and spin ;(0) ) − 1 ≡ P ≡ P n E n/ ) )  E (  E E n = ≡ P ( ( is the anti-de Sitter energy operator and These elements are the initial data for P 1 2 ) n   E E 20 := ), P P ( Turning to the spaces The reference states can be taken to be the scalar ground states represented by the scalar-field mode in spacetime with energy n n E More precisely, in [ As a consequence of the action of parity on the oscillators ( ( e 2 representation of P P 19 20 n / versa. tinguish them from theangular rank-1, momentum biaxially-symmetric projectors twisted projectors that map the projectors associated to particle states to by transforming singleton states into anti-singleton states with opposite energy, and vice- of the second equation inthe ( found by Breitenlohner and Freedman [ of the particle states in our gauge, as the interactions involve Thus, under the twisted-adjointizations action of ( rotationally invariantwords, the modes twisted-adjoint of action of an on the tensor productmodes, of two in super(anti-)singleton accordance states, with that the gives Flato-Fronsdal rise theorem to [ specific scalar from which it followsenergy that eigenvalues are they given are by rotationally invariant and that their twisted-adjoint where the notation These projectors obey where 1 shall limit the expansion of Φ P with energy projectors on the scalar and spinor (anti-)singleton ground states, respectively, JHEP10(2017)130 ], the (4.13) (4.11) (4.12) a (her- 2 , , 1 , in terms B , however, 0 α b ηE V 4 − J, iB e generalized pro- n =  and the gauge func- K 1 0 ) + 1 − , η η ,... 2  = 0 , and hence the full solutions  of the Cartan subalgebra of y , πi y dη 1 κ 2 e K  ) Weyl zero-form component (see ? κ ε , and therefore Ψ and ( ? 0 = s C )] I and n is an angular momentum, E ε ( ]( 2 K n J 1 We recall that, as studied in [ 1+ P − ]. Indeed, following the same reasoning as E, J, iB, iP . n 3 E, , 1) = 1 ) admits a generalization to six different in- − J ) alone. , where = 0, as dictated by the spherical symmetry and – 29 – ; ) = [ } K r E E ( ( ( n ) ), with different physical meaning, labelled by pairs ) = 2( (4) orbits give rise to linearized solutions spaces given ,K P  ( e ? π E ) K hs ; n D (8 K e 1 ( P K ( n − ≡ ) − (1) n P E, J, iB, iP ) can be realized as [ )  L ( n { ) already at second order, and hence in the spacetime gauge  ˙ e ( 0 α E P D 4.7 4 ¯ b V D − , e 0 α E? ε b a (hermitian) translation. Thus, restricting to symmetry-enhanced V 2 1+ P − n 1) ) given in ( − E ( for a quick review of Petrov’s classification). n , their fully non-linear completions, which are soliton-like states, contain twisted P ) are eigenstates of one-sided star multiplication by A ) = 4( K to create a branch of moduli space that is smooth at the origin of spacetime, which ) taken from the set . To this end, let us start by recalling that the rotationally-invariant supersingleton ( E n ( C E, iP L Thus, starting from a linearized particle mode, the star-product interactions generate n P = P (4; of projectors and twisted projectors,jectors which we shall referprojector to collectively as the can be given by expansions over the 4.2 Regular presentationNext, of as generalized anticipated, we projectors shall expand the initial datum Φ The corresponding twisted-adjoint by expansions over lowest-weightK spaces that inprojectors general (that necessarily not appear unitary. at higher orders). In In the the cases cases that that equivalent solution spaces of mutually commuting andsp normalized elements mitian) boost and solutions, the construction sketched above can be repeated for the more general projectors conformal field theoriestowards at the zero conclusions. temprerature; we shallRemark comment on on more this generalsolution open Killing space problem vectors. based on fields once the gauge functionticle is and switched black on. hole We modes interpretVasiliev equations. the as perturbative This a raises mixing manifestation the of of issuetion par- the of whether non-locality one of can the fine-tunecould Φ interactions correspond in the holographically, via suitable generating functions, to three-dimensional space modes give risemodes to exhibit the fields typical that singularitythe are at generalized regular Petrov everywhere Typeappendix in D spacetime, structure the in black each hole spin- black hole modes in ( above, one can show that implying that the twisteddetailed projectors nature of give rise the linearized to fluctation static fields, solutions. we Concerning recall the that more while the lowest-weight generalizations of Schwarzschild black holes [ JHEP10(2017)130 ¯ Ψ see 21 (4.19) (4.20) (4.15) (4.16) (4.17) (4.18) (4.14) viz. ]. 3 with positive , ] to solutions . n 1  ) n P ¯ y P 0 n e ν amounts to [ iησ + y − κ n y e P ( n 2 ν δ of the generalized projectors.  , n ) follow by associativity; ,...   2 n 1  e , P 4.18 1 , , + 1 − n X  e ν n n η η , , . As for the twisted projectors, the are abelian. We shall comment on alternative )–( | = e P P ˙ n n  β + n n ¯ e y | m m P P ˙ n β = − − 4.16 α πi P n/ dη ) nm nm n, n, y 2 n 0 δ δ δ δ ν ) σ ε (  := ? κ ( i = = = = C ∓ – 30 –  ε ,... m m m m I α n 2 regular presentations e e y e P P P P  ε P , ) and performed the star products before evaluating 2 n 1 ? ? ? ? 1+ X e ν  n n n n − = e e P P P P n + 2.45 n ) n ) as the − P ( n π Ψ = ) encircling ν 4.14 ε ), from which eqs. (  ( = 4 C ,... )), we note that the fact that the product of two twisted projectors y 2 over projectors and twisted projectors, for which we shall use the 4.15  0 , ) and ( 1 ? κ 3.30 X  ) ) which are divergent as -product algebra containing an ideal spanned by the projectors, ¯ y = ? E 4.13 0 n ( ] for a proof and a precise prescription for the contour. For this reason, we in eq. ( iσ n 1 = ∓ P 0 n y ) are generalized Laguerre polynomials and the contour integral is performed ( Φ 2 x := δ ( n variables, since that is what the star-multiplication with e (1) n P y L Performing the auxiliary contour integrals first, on the other hand, one encounters star products As the nonlinear corrections are given by star-powers of the adjoint elements Ψ and 21 between two perturbative schemes towards the conclusions. following notation: i.e., 4.3 Explicit form ofWith internal the solution purpose insuperposing holomorphic of particle gauge generalizing and the black holeof black-hole-like modes, the solutions let initial us found datum examine Φ in in [ more detailed the expansion is an ordinary projectorfull manifests equations, but the rather fact their completionhigher that into spin full free solutions black scalar includes hole spherically modesmassless symmetric modes. alone scalar It do and is not higher alsoto solve spin interesting the the black to hole note modes that is the just relation a between Fourier free transform with respect and negative appendix F in [ shall refer to eqs. ( (see for example eq. ( where we have used thethe property contour ( integrals. In this way, we achieve orthonormality between the The twisted projectors aresubalgebra not of idempotent, the but rather the generalized projectors form a around a smallcorresponding contour presentation is given by where JHEP10(2017)130 )) (4.25) (4.26) (4.27) (4.28) (4.29) (4.30) (4.21) (4.22) (4.23) (4.24) 3.30 , i k ) = 0, and the = 0, i.e., the absorb all the . n n (see eq. ( ) Γ n k e P n,α 0 , defined by ( n P α − n e n e ν V b V e ν n ) must run symmet- , 1) − k n − ν ν 4.19 and + R = ( = ) that n and the n ν n ∈ ) , ν n k P ( n )(  n P 4.27 ν n ? n , . e Γ ¯ y P ) , ¯ κ 1) 1) k . , µ − y , n ( − − n n n k k e ν n e ν that appears in ( ( i n n ν ν µ . One has k e e ν ν ) impose the following restrictions on ], with = + n 2 ) ∆ n n | i ?k n (∆ n ) 2 n − − , Γ k n e (1) ν , κ e n ν ν = P ( 2.33 n | 1 are cancelled by zeroes in the numerators. − e ν ) e ν n e ν k n − + + k n ( n − ν ) n ν + 4 = n P 1) 1) −  2 ) − − ν n ) – 31 – k , k k ( n − ( ( ,... n n + Γ ⇒ n + ) = 2 ν ν ν ν ν n n  n n n , = 0 it follows from ( − e ν ν − P ν 1 (∆ , = X ( ( ) ν  n n k = = p ( = n e ν ν − + (1) ) ) n n ν n k k k . ν n ( ( := n n ) ν 1) = n e ν ν n n ) to write − , π )( − e ?k ν † n ) + Γ Ψ n = ( = 4.18 n P + Γ ∗ ∗ n n and Γ − such that e ν ν )–( n ν n ν = ( n − + are homogeneous polynomials built out of ν n (∆ 4.15 n ) P h are constant deformation parameters, the k ν − ( ) it follows that the sum over the particle states ( n ( = 0, such that every particle mode is accompanied by its counterpart with n n e ν n h ν Γ n e ν ) = n 1 2, by the recursive relations n 4.23 n Γ +1 := e ν k k P and ≥ ( 2 2 and n ) ¯ π k k ν ( n n π = = ν ν ) ) k k ( ( It is now possible to evaluate the product Ψ n n e ν ν -dependence and obey We note that the zeroesMoreover, in for the denominator every Γ problem reduces to theparticle sector one disappears. already solved in [ where ∆ The solution to these relations is given by and, for where rically around opposite energy. explicitly in terms ofprojector algebra the ( deformation parameters. To this end, we use the generalized Thus, from ( Y As a consequence, thethe reality deformation conditions parameters: on Φ ( where JHEP10(2017)130 (4.43) (4.36) (4.37) (4.38) (4.39) (4.40) (4.31) (4.32) (4.33) (4.35) (4.41) (4.42) (4.34) ) among the , , z z ) in generalized w w 4.30 ˙ 0 α 1 1 ¯ b +1 +1 − − V t t t t , i i 0 α e e , , , b V 1 1 ), we have defined the  ,  − − n n k k , e P e n,α P )) )) 4.23 ) = 0 ) = 0 1)! 1)!  e and ( ? 2 2 V ) yields the two conditions ? are given by n 0 ) ) − /t /t n,α n,α − − e P Z e 3.4 V − Z k k ( ? ( e ( n,α ( ( V . e ( π ) = V n,α n π n,α n,α (log(1 (log(1 ν ). Using the generalized projector n n,α ¯ e e e V V n,α V , e e ν 2 2 V e V + and . in eq. ( ( 3.6 + +  + n + ∗ n n n ν n n n ) dt dt , e e ν n n,α + 1) + 1) )–( P n,α P n P P ∆ P , e e ν V n,α n t t e ν V ? ? ( ( e e ( n ? ν V n,α 3.4 = ) ) n − 1 1 e n,α , π V 1 1 and + ). We note that the relation ( ν e − − − Z Z V n := n,α ( ( e ν n Z Z n – 32 – n,α ˙ V iϕ n n,α α n e ) ) P 2 n P − ν  k k V 4.29 n,α n, ], is to first expand Φ n e iϕ ν ( ( − n n )) + V ¯ 2 ∆ 1 ν V V − e e ν ν ,... )) + e 2    − k k n,α = =  Another way of obtaining the internal connection, which , n,α −   ,... ,... ,... ) = 1 2 2 2 V V b b 2 2 n,α X ) and (  ( (    n,α n,α V , , , − − = n,α π π 1 1 1 − − n X X X ), the internal connection takes the form V e V V    ( + + 4.28 = = = = π     − n n n 2 2 0 3.30 α n,α n,α / / k k b V = = = V V 1 1 n,α . Thus, the coefficients ( ( 0 0 0 α α ˙ V   α n n Φ b V ¯ b V e ν ν ¯ b V ), the coefficients of =1 =1 ∞ ∞ k k X X ) into ( given by ( α α ) k 4.18 ( iz iz n 4.24 e ν )–( 6= 0, and where, recalling the reality conditions ( = 2 = 2 viz. n and e ν 4.15 n,α n,α ) e k V V ( n ν Inserting ( provided phase factor respectively. They are solved by and then insertalgebra these ( expansions into eqs. ( Alternative solution method. is closer to theprojectors, procedure followed in [ deformation parameters translates into This completes the solution in twistor space, i.e., in holomorphic gauge. with and analogously for JHEP10(2017)130 ), )– . 0 α b V 3.25 (4.53) (4.54) (4.50) (4.51) (4.52) (4.48) (4.49) (4.44) (4.45) (4.46) (4.47) 4.48 as soon n,α V , . ) and ( z z κ κ ] and references n n 3 3.24 e ν , b ν b , . = 0 does not imply z z ; as for the fact that κ κ αβ αβ k n n n n   ν ν -space connection e = 0 = 0 ν i i 4 4 , b ν b ? ? Z = ] ] , − − ˙ ˙ ) α α  , αβ αβ k  = = (   n n, n, z ] ] ν i i 4 4 n,α − − κ ] ¯ e ) as V V = 0 ¯ e V − − n,β n,β  , [ , ] n n n e e V V e  ν = = Γ , . [ n,α 4.42 n,α ? ? ] ] n,α e bM V Σ e V α α  ˙ [ [ α n,α n,α n,β n,β = 0 and − ) gives rise to the conditions e n, n, V V e + [ V − + [ e ˙ ˙ 1 V V n,α α α e ? V ? ? ¯ ¯ ]  n 3.6 ∂ ∂ − n,α ] ) and ( , ∂ ˙ ? + α ˙ e α V α V iϕ [ ] ), which is equivalent to ( ] 2 αβ n, α nonetheless provide a source for n, [  n, form a closed subsector, whereas the particle + [ n,α − , leading to the holomorphic Ansatz satisfying ¯ ¯ e i 4.41 V V to vanish, one obtains V n,β n, e 3.5 2 Σ , , e e V V – 33 – n n,α = 0 = = 0 = n,α , it is possible to use the gauge freedom to set to + n,α − − + n,α e e ? e V P V ] n ˙ ˙ n,α n,α ] V + α α V ], with , e ν α ] = V V [  n, n, , 2 n,β n,β ¯ ) = i ? , V n n, ¯ e n,α V ] n,β − i . Next, eq. ( = 0 reduces the deformed oscillator problem ( and 1 + [ + [ e ν α V α ] ˙ V  − α ∂ [ n,α ?V ∂ n  n n [ n,β ¯ ?V ?V b α V n e α ν ν n,α n,α P [ z (Σ e Σ ν α α e V V [ [ ∆ , n, π ˙ ˙ α α ] n, n, V := ¯ ¯ ∂ ∂ e  − V V [ n,α ] ] + − − Σ  1, as we shall examine in detail further below. Thus, introducing ] + + [ n,α h ˙ ˙ ] ] α α n,β Σ n,β n, n, ¯ n,β n,β V V ¯ k > e ?V V e V V α α = 0, on the other hand, the black hole sector resurfaces through the [ α α α α ∂ [ [ [ ∂ ∂ n ∂ ∂ n, ν e V + ] n,β vanishes for e V ) α [ k ( n ∂ ) to the one already solved in [ ν We proceed by defining the normal modes as follows: 4.49 in terms which one obtainsn a set of two decoupled deformed-oscillator equations for every a particle mode necessarily turns on the black hole sector in the therein. Setting nonlinearities: even though theright-hande side, first the equation quadratic has termsas no in one curvature goes deformation beyond termthat the on linearized the approximation. Indeed, setting It follows that themodes black hole do modes not. of ( Indeed, setting black hole modes obey both the linear and the non-linear equations, see [ which can be rewritten using the relations ( and requiring the coefficients of zero any non-holomorphic terms in Finally, inserting the expansions into eq. ( Solving perturbatively in powers of Identical considerations hold for JHEP10(2017)130 ) , 1 = 0, 4.56 − (4.65) (4.61) (4.62) (4.63) (4.64) (4.55) (4.56) (4.57) (4.58) (4.59) (4.60) k k ] given λ )] 1)! 2 64 /t − k ( , , z z z w w [log(1 w 1 1 ) with parameter 1 k +1 +1 − − . Choosing +1 − t t t t t t z ! i i i . ] w e e C e 3.29 ) 1  [ 1 1 n +1 − u 1 2 t t − − i − k k − e k bM , 0 )) )) )] 1)! 1)! 1 z 2 2 1)! − , tt 2 w − ( ] /t /t − −

k 1 δ /t −   [ +1 − n ) )] k k . t t  k 0 1)! 2 ( ( n 2 i t ( 2 n ) ν e ( / k /t n − bM Γ 2 ) , see appendix 1 ∆ (log(1 (log(1 [log(1 t Γ h k -product algebra; fur further details k (  2 2 ( − ] ) ◦ 2 λ − t   ( [ n 1 1) =1 n ∞ 1 [log(1 f k X dt dt −  h + 1) + 1) dt 2 − + 1) 2 0 ν t t k = t t ( ( ) ( ( ] dt + # 1 1 δ 1 1 − 1 dt 1 dt ) 1 [ 1 + 1) n n + 1) − − 2 − − ν t ) that t Z Z – 34 – Z ( ( Z M ( /t ) ) ) = k k k t 1 1 2 1 1 1 ( ( dt n n −  − + ( 4.41 )( e -product problem is solved by ν ν 1 ] 1 Z b Z 2 ◦ k k := −  log(1  k i [ n α ] Z −   ] k n z f   ) n b b 2 2 ν  [ ] n [ , b n 2 ◦ Γ 2 − , the ∆ ) ] − − [   n M = 4 t − ) := n  2 ( [ bM n u ] / M k k f , (  1     p )( ( 1 + [ n,α   ) 2 2 2 ) act as projectors in the ; 2;  t 2 −  Σ t / / k k h σ,k ( 1 2 / ( k ] 1 1 k k ) λ ◦ 1 =1 ) ) ∞ "  k [   k X ( 1  1 j δ =0 [+] ∞ h [+] n n n n F ) combined with ( k X ( =1 =1 ∞ ∞ k e . For each ν 1 Γ =1 k k ∞ X X t ] M ! M k X 1) ( k ( α α α  1) + [ − ) + h  n 4.52 α ( t 4 iz iz iz ( − iz ] × × ) obeys the integral equations bM t  t ( [ ]. The solution reads = 2 = 2 = = 2 ( δ q − ] ) := t  [ ( 67 n,α n,α , n k e ) = ) = ) = V V p f t t t ( ( ( 66 ] ] ] ,    1 [ [ [ j q identified as 1 + f and it follows from ( where as well as the explicit expression for the coefficients Using the above integral representation, thereproduces commutator the of singular two deformed source oscillators term ( using the limit representation ( where where These equations can bein solved using [ the refined version of the method found in [ with complex deformation parameters JHEP10(2017)130 1; − (4.67) (4.74) (4.68) (4.69) (4.70) (4.71) (4.72) (4.73) (4.66) = ). We t 4.33 ) necessarily )–( 3.5 4.32 , , )). More precisely, z = 0 holds identically,  κ ˙ on the right-hand side α n 4.18 0 . Defining the reduced n,α α , e ν y e b V F κ )–( n,α αβ ? n,α  , V e y , V 4 ib κ n  , 4.15 ) are real analytic away from ] iϕ − . n,α + 2 , ,  n,α e [ V ˙ − e = 0 = n α V − e ] ] n,α n ¯ = 0 Σ viz. = 0 V , iϕ = ] n,β n,β ) = 0  −  ) = n,α [ source the − − e ? n,α e n α e V V n,α n,α e V ˙ n α n,α  V − ? ?V n e P V n,α e V − ( α α e V [ [ e ν n,α 6=0 ) + n, n, V X n e e V V  – 35 – , ∂ ) + n,α i = + + − 2 0 , π ] ] α . Moreover, the condition V n,α b ∗ ( , V = 0 that = 0 − − ) n,β n,β 0 π n,α e V α n ( e V arise from the singularities of the integrand at V b ( Φ z e ν n,α n,α n ( 0 π α ) now only has components along the twisted projectors, ?V ? ). − V e V ν / Z n b , ˙ α α := D α n e [ [ ν = n ∂ e ν ] n, n, 3.24 3.30 ) = P  V V [ n,α = n n,α e ν Σ n,α n + + − V ] ] V ) reduce to iϕ ( 6=0 2 X n π e n,β n,β e V V 4.51 = α α 0 [ [ ∂ ∂ b Φ ) and ( , i.e., in which the Weyl zero-form contains only scalar modes. In this case n 4.50 ∀ . As already noted, even if the zero-form is expanded over only projectors (and ], where singularities in ˙ 0 α 1 , ) requires the linearized internal connection to contain only twisted projectors, but ¯ b V Proceeding, it follows from 3.5 = 0 = 0 [ n where, as already noted,deformed oscillators quadratic terms in The deformation term in eq.i.e., ( eqs. ( where the phase factor provided that the internal connection is holomorphic, with the solution the higher order correctionsproduce to both the types commutator of oneven- projectors, the and left-hand as odd-order side is correctionsand of evident to twisted ( from projectors, the eqs. respectively. internal ( connection are expanded over untwisted idem no twisted projectors),and the twisted internal projectors. connection Atof must the ( be linearized expanded level, the over presence both of projectors In what follows, we giveν explicitly the solution informulas holomorphic simplify gauge and in the theorders backreaction case becomes mechanism in more producing transparent which black The hole resulting modes Ansatz for at the higher internal master fields reads remark that the reducedZ deformed oscillators (Σ see further comments below ( 4.4 Black-hole backreaction from scalar particle modes which coincides with the result that we had previously found in eqs. ( JHEP10(2017)130 and n (4.75) (4.76) (4.77) (4.78) (4.79) ]. Whether 116 = 0. Solving (i.e., the two n y ν ? κ , viz. , z , n z w n w P 1 . 1 +1 − +1 − t t t t = i i e e   , n , is non-vanishing beyond the  e x P  evaluated at 2 ) 2 ] − t z t = 0 n,α  κ [ n | ] V ; 2; log n log  M [ n,α that have energy eigenvalues 1 2 e | ν | | n  Σ b n m 2 e ν ˙ α 2 e 1 ν | | P b ∓ F b 1   (1 The latter, in fact, is only responsible for , ∂ +  − and ] F αβ 22 F n   [ n,α 2 i 2 P x 2 Σ − − – 36 – dt ; 2; dt + 1) + 1) 1 2 = t t (  ( ) = ? ] 1 1 i 1 space. Clearly, sufficiently large changes of basis, by 1 1 ]  F − [ n,α − 1  Z [ n,β Z Z  (Σ Σ | n , π 1 2 e ν n 2 ] e ν of the regularization procedure used to define the underlying 2 ib |  [ n,α ib α Σ for further details). ] := z h not α x [ z −  4.5 = = F at the level of the Weyl zero-form, particle modes are encoded in the n,α n,α ]), and e i) V V 11 , and black-hole ones in the twisted projectors 9 . n n e P ν = 0, one obtains k λ We note that, working within our scheme, the above effect is a consequence of the We stress that the appearance of black-hole modes at second order in (classical) pertur- Recently, working in a different basis, and imposing specific gauges and boundary conditions, it has been of opposite sign. Instead, the appearance of black-hole modes is purely a consequence 22 of the fact that: projectors shown that Vasiliev’s equations atthis quadratic scheme order can correspond bethe to extended current quasi-local to scheme, cubic a remain vertices fully open [ non-linear problems. quasi-local deformed Fronsdal theory, and be related to spacetime non-locality induced by thefor twistor example, space [ non-locality ofassociative the algebra star (see products (see, setting to zero the star productsm of projectors keep in mind thatinvariant the functionals actual thereof, observables are andperturbatively not we the produced expect master black-hole an fieldsbuilt agreement modes themselves, from at but provide particle rather that additional polarization level, tensors. corrections noting to that observables the to Vasiliev’s equations constructed bymaster working fields with that a facilitate specific the setas imposition of well basis of as functions boundary in for andmeans the the gauge of conditions internal in large twistor spacetime gaugeorganization of transformations the or master field fields redefinitions, and hence may the affect above the phenomenon. perturbative However, one should One can indeed see, as expected,leading that order; the it black hole contains sector contributionsparameter that are of positive even orders in the deformation bation theory, starting from particle modes, takes place within families of exact solutions were we have defined the even and odd combinations these equations using the methodtaking based on integral representations spelled out above, and where the deformation parameter indeed corresponds to one arrives at the decoupled deformed-oscillator equations for every JHEP10(2017)130 (4.82) (4.83) (4.84) (4.85) (4.80) (4.81) , ) 24 ¯ y , , 0 ) 23 n − iησ the Ansatz we use P − + y ii) ( , the reality conditions . The latter hold with n 2 n δ P  also its negative energy − ( n n n iii) P n e P −  P e ν ? = 1 n n e ν + 1 − , , − n,α η η e V , n P y ηE ) = −  ? + 4 n P n − n P n ? κ − πi e dη n P − 2 P  n e ν − ? ) n e ν ε  e ( P + 1 C n + n,α and n e ν I V n + 1 − P n  ε − n + η η P 2 e ν P – 37 – 1+ n ,...  n = 2 − P = e ν  n n ( n , ) πi 1 ν pt dη ? P X 2   − ) ? ) ( = ε n is proportional to n π ,... ( n − 2 i C P α  P 2 I , b Φ = Φ V = 4 n 1 X ε − −  y 2 e ν = α 1+ n z − ? κ + n ) = = n 0 1) E P 0 α ( Φ n b − S n e ν P . These facts imply that, starting from, say, a single particle mode of = ( , which means that in the deformed oscillators the particles are represented n 2 y ) := ) = 2( − ? ) P E E y ?κ ( ( n n and its corresponding anti-particle mode, e ? κ P P n (Φ To this end, we first summarize the internal master fields as follows: While the proof is more straightforward with the solutions cast in factorized form,We we remark would that the like fact to that the connection on a non-commutative symplectic geometry can be shifted 24 23 stress that theproducts latter among is master not fieldsnormal-ordered necessary in form. in our solution any space, way, can and be everyto proved result, a equally deformed in well oscillator in particularwhich that factorized are the belong and gauge to finiteness invariant in an observables of fully adjoint that star section have no is analog important in for the constructing commuting open case. Wilson lines, with Let us demonstrate that theforms enveloping a algebra subalgebra of of the an internal associative deformed algebra oscillator with algebra well-defined star product. form of the projectors,the obtained regularization by is first only evaluating responsible the for contour setting to integrals. zero In the other other4.5 words, two mixed star products. Embedding into an associative algebra corresponding, as recalledresulting above, from the to star black-hole products or states without the in regularization, the that is, deformed one gets oscillators, the thus same result also using the unregularized energy the second order correction to reconstructs the deformed oscillatorspowers (hence of Φ the gauge fieldby generating twisted functions) projectors in andforce star the the Weyl black zero-form holescounterpart to by contain projectors; for and every particle mode types of modes are connected via a twistor-space Fourier transform); JHEP10(2017)130 , 0 )– ρ viz. , thus 0 (4.94) (4.86) (4.87) (4.90) (4.91) (4.92) (4.88) (4.89) =0= 4.15    b ρ j ¯ A

spanned ? ) ) , then − ¯ z R ρ κ + (¯ 0 , ρ , , ∈ R ? i 2 , z z 0 n + ) w w ρ ?j =0 ). From the fact 1 1 ) ρ m 0−

z , +1 +1 − − ρ s t t t t , T g ) κ i i =0= + + ( 4.29 ρ m e e ρ z =0

i ··· 2 ? ρ ) 1 1 −

i 1 ¯ ρ )–( , T f − − ) − − ? s ) ρ + k k − ) + p ρ z ¯ + z y − ( α 0 )) )) − ]), it follows that the asso- − z 1)! 1)! ρ − κ 4.28 2 2 ∂ ˜ ρ t 1 ρ i (¯ + 2 ( ρ − /t /t − − − ? δ + + − ··· ) − k k z 0− ) z − ?i , ( ( 1 m ρ ρ + z ) + ( ) α ρ + y ρ + n ( ∂ ρ z +˜ κ (log(1 (log(1 ) g i 2 ( − t i 1) +1 2 2 z ( m ˜ − t ? − p f + t , . . . , s + e ) ( ( z 1 z ) (see also [ ( ) z s dt dt − ...α 1) are given in ( T ) + 1) + 1) (¯ j ¯ ( ˜ 1 ρ i ρ j ¯ +1 − t t ) ( t ˜ t C.5 − m ( ( k i,j, ( ¯ variables. Indeed, if e ( f n i,j, − n,α ¯ ( i, ) = 1 1 1 1 z e n i, ν ) f m n ρ, ∂ i − − ) + m +1 ). The star product of two elements in ) belong to the ring product algebra ; ˜ V t ˜ ρ ρ t Z Z t 0 – 38 – 0 ( ds ( =0 t e ( of the form p ) ) T g ? ∂ p ( X 1 ). We can rearrange p ( k k 1 and ) i ))( ) ; I ( ( n n +1 ) 0 − ˜ t t Z ◦ 0 z ) e ν ν m ( Z ...α ρ ( e k j ¯ ) ( j C.6 ¯ 1 k k ( j ¯ n )∆ ) ) := ∂ m ) ν )   ) i,j, ¯ ; ··· i,j, 0 ρ ¯ i,j, ¯ ρ 0 n,α i, ( b b n 2 2 ρ n i, ( t 1 i, ( T f f ∂ ( ∂ , . . . , s V ( 0 − − ∂ ∆ ; ; ds 1 t 1 ; t , 0 s ( 1 ( 1 t )∆ ( I j ¯ ( − ) dt     + 1 =0 0 f ρ j Z ¯ ∆ 2 2 ( p t i,j, ¯ X ) / / k k 1] of the form ∂ n i, 1 )∆ i,j, ¯ 1 , 1 1 ; I ) m i, t − 1 ρ ∆ X s   ( ) := Z ? are defined in ( t − ∂ ∆( ) ; =1 =1 =: ··· ∞ ∞ )(  t := k k X X ) being a polynomial in E 1 ρ m ( j ¯ α α s m +1) 1 n ∆( ˜ t )) (( iz iz i,j, T f ¯ 1 2 P n 2( ( i, T 0 and ˜ V n 0 C.4 = 2 = 2 dt ) (4.93) X , . . . , s 1 tt 1 z 1 ) = is a finite interger, and − (    t s 0 forms a linear space under addition and multiplication by polynomials, n,α n,α = ( Z 0 e )( V V ˜ t p m := dt m f ?V R ), and using the self-replication formula ( 1 0 1 ) b − A z T f Z ( ( 4.18 p = where involves holomorphic star products in V with and t where by real functions on [ are defined in terms of where (see ( where the deformation coefficients that the generalized projectors( form a separateciative algebra star generated product by the algebra, internal as master in fields is eqs. a ( subalgebra of and JHEP10(2017)130 ; , and , (5.1) =0 (4.95) ρ  }

) , and ) -gauge · ) 0 ) t ( q L ρ ( + ) 0 ( β p z p ρ ) is a local ∂ ( γ − ; hence the 0 ρ ∂ ...α t 1 − ··· Vasiliev gauge α x, Z − ) ··· ( 1 z f ρ as to remove all ) ( e β L { + 1 ρ and ρ ∂ ( γ 0 Y ) that activate the t + ∂ p 0 − α q z ρ β + ρ z ··· 1) x, Y, Z , ), where ··· 1 ( − t α 1 b g ( β ρ ( 1 only arise in the black hole ) ρ x, Z = Ω , as to give rise to a nontrivial t i (  +1 ( t 0 > e X ) with that of L q p e (0) ρ ( s α ? b U ) Z ∂ ...β ) t 1 ,β ··· 0 ))( , these functions implement large gauge p ) ) x, Y 1 ρ 3 ρ ( , ( ( -gauge, the internal connection becomes α ...α L (after Lorentz covariantization and setting α ∂ 1 L ) Z t ρ, ∂ I,α W ( )( ) = = I are linear combinations of f 0 0 p ), that takes the solution to the – 39 – , which facilitates the perturbative construction spacetime provides a vacuum configuration for ∆ (0) =0 o α 0 0 0 ...γ X ) ◦ b p q S 1 0 4 t ) ,γ ( x, Y, Z ) 0 P 0 ( ρ q . Thus q ( x, Y, Z b L ) (4.96) AdS ( ∂ · R =0 ...β , ( 0 0 ...β b 1 H p p 1 I )( t ,β ? ( I,β 0 P 0 (∆ The = 0 p )  f ) ) ◦ ...α (0) ρ 1 dt under the star product. + 1 ( p b x, Y Φ 0 t ( I,α , linearized Fronsdal fields with 1 0 ...α A L 1 ρ, ∂ 1 − f 4 ( ) = Z I n ) 0 I,α ρ for generic points in ( ) = f I ∆ ( X ◦ Z 0 0 and =0 ρ, ∂ ) p 0 ) and the dependence of the master fields on ; X p 0 , respectively, with coefficients given by polynomials in ρ ) = t (

x, Y, Z b z ( ) U ∂ ) =0 ( ( 0 t I 0 t ( 0 0 p ( b ( I G X p 0 p,q p vacuum solution. ∆ ?V ), that takes the solution space to a gauge that we shall refer to as the ...α  ) 1 ...α 4 , that we shall undertake at the first order. We also note that, due to the nature of 1 z := ( b α = 0). We would like to stress that in the I,α 0 H x, Y ( V f f 5.1 Gauge functions AdS Vasiliev’s theory, given by real-analytic in of the Ansatz in section sector; we shall discussthe the Conclusion. prospect of switching on these fields in the particle sector in singularities away from the center ofof the black the hole Kruskal solutions, coordinate leading system tofunction for a the gauge reminiscent Schwarzschild solution; andwhere iii) Fronsdal the fields Vasiliev gauge arise asymptoticallyZ in but that can nonethelessfields. be In what useful follows, inL we order shall consider to three remove gaugeii) unphysical functions: the singularities Kruskal-like i) gauge form the function AdSLorentz the gauge transformation function that aligns the spin frame in In this section, weconnection shall introduce differentspacetime structure. gauge As functions already stressed intransformations, section altering the asymptotics of thetransformations, fields, as which opposed to represent small, redundancies or proper, in gauge the local description of the dynamics, showing the closure of 5 Spacetime dependence of the master fields where   new functions remain elements of where ∆ JHEP10(2017)130 , (0) A e (4). = 1 (5.7) (5.8) (5.9) (5.2) (5.3) (5.4) (5.5) (5.6) hs (5.11) (5.10) = λ ), which (0) L , Y W . ( 0 2 =0  x Y

= − Ω 1 ] is a map (0)   b ˙ p β 86 ¯ , y 2 ∂ , := ˙ 85 are given by ∂ α , The metric can be given ¯ y a ω , ∂ 6= 1 x ) ˙ , α 2 β α , ) , h , a x, Y 2 b β ∂y ) ( σ x ∂ Y α C a β (4) or its minimal subalgebra , x α ∂y ?L = ( 1 ?L, . dx 4 , L ) via ˙ )  α with invertible frame field. The result- ) hs R R α αβ i Y | Y L = SL(2; L ) ( ( X ∈ 0 / Y . ab ) ( a 4 ? f σ R ?  ?L f = Ω ) of ( ) = 2 ) 1 2 ˙ α x β , x ( ˙ − − (0) α AdS ˙ ) = . This coordinate system provides a global cover α – 40 – h L ¯ ω ?Y αβ x, Y α Sp(4; ? dL , , x ab Y − ( 1 x L ( = η . Correspondingly, in the notation of appendix 1 1 = 1 is taken to be two-sided, after which it can be 2 ˙ − α b (0) L αβ ) − = − α 2 R . A vacuum gauge function [ L 2 f ω 2 A )

L L x dx λ ( x a R → a (0) αβ dx (0) := σ − :  ( b 4 dx L ) := 1 2 α L (1 Y , Y − := ( = where it obeys , ω L 2 = =0 f a Y a 2 (0) and the canonical Lorentz connection dx

x dx Ω ds ˙ α ˙ )-valued flat connection on W α α and ¯ y ) R R ⊂ X a 2 ∂ ab σ ∂ α (4; η ( b 2 ∂y sp x − a h iλ x − := = = 2 belongs to the bosonic higher spin algebra , where , provided that the surface 2 ˙ ˙ 0 α α 4  x (0) α (0) α (0) e e ω AdS we have Defining where of identified as the boundary; forric relations to global embedding coordinates, coordinates and see spherically symmet- appendix are globally defined Killing parameters on Vacuum gauge functionon in manifestly stereographic Lorentz coordinates. covariantas form follows: in stereographic coordinates in units where it follows that where Weyl ordering is assumed on both sides; in particular, we have where Writing and defining the matrix representation The Killing symmetries, thatpreserving the is, vacuum, the have parameters globally given defined locally higher by spin gauge transformations introducing the inverse AdS radius defined on a region ing vacuum values of and where Ω is the JHEP10(2017)130 ), Z C (5.18) (5.19) (5.20) (5.21) (5.22) (5.12) (5.13) (5.16) (5.17) (5.14) (5.15) (4; sp ) generator ∈ R , ) . M ˜ L, 0 L β 0 E ? d ( α ) ) 1 0 C 0 − 0 ˜ L (Γ . SL(2; 0 . β a ! β /C h ˙ P ) ˙ β L , β , , as we shall exemplify further a 0 ˙ = Ω + α x α C α C δ h h x ix (0) 1 + , α ) ) 4 ω − L ] )(0)

b ], and as we shall review below, this L 1 1 + ξ x ξ x 1 ( = = SL(2; 1 , 87 exp b , ) U b r ) that commutes with H, < n L h → αβ 1 . ( )(0) C h 86 − b 4 b L 2 , H sinh(2 2 ( , though the internal connection does no cosh(2 L? x ) 1 R 1 + = 0 ], we define 66 β with one adapted to the Sp(4; : β > β 1 ) : ˙ ˜ α n X x α tanh L, 4 b ) = δ G ¯ x Z ? y ,E ˜ 1 2 ( ? R α L ) = ) ) – 41 – a S α − α , ω ( L ∈ αβ ) α P ξ x y ξ x Y,Z 2 a a | E . As shown in [ Z = 1 + h ?Z β x x L n , = ( = Ω 1  b ) − Y iξx P H b − G α Following [ Z = (4 ˜ =0 | sinh(2 L cosh(2 )(0) Y ? Z x b | L 1 8 ( R

= ( ) := (1 ˜ b Provided that the internal connection is real analytic in L L − ( αβ = ξ W ? , the Vasiliev gauge function )(0) = = exp β c ) M b L α ( α Y L | b L S x ( X × Y L may in general contain singularities in ?E?L ) , n ) denotes the subgroup of SL(2; 1 ( = 0 and − ) := b M H L = 0 ( is to align the spin-frame on =0 ) Z C = ˜ | L )(0) ˜ L L b L x, Y, Z is a perturbatively defined field-dependent large gauge transformation defined by ( ( E SL(2; ? d b b Φ L b C H 1 . In this gauge, the vacuum configuration is given by − 4 used to construct the projectors ˜ L L We note that and stress that below at the linearized level in the particle sector. where as longer obey the Vasiliev gauge condition. Vasiliev gauge function. for generic points on gauge choice removes all singularitiesAdS from the gauge fields away from the spatial origin of implying that still and The role of E Kruskal-like gauge function. where Its matrix representation reads the vacuum connection can be integrated on and expressed in terms of the gauge function [ and JHEP10(2017)130 (5.31) (5.26) (5.28) (5.29) (5.23) (5.24) (5.25) (5.27) . ; (5.30) ˙ β Weyl tensors ) , E  ) yields − ( ˙ . β y ¯ u β  ˙ ,... α ) β , ) 2 L 5.26 α L ? κ y E , v v − ( ˙ β ( L , n ¯ u y 1 ? κ in ( e a P = 1 − αβ y ). In terms of the global + , , L n y ) =0 ˙ -dependent eigenspinors s β ˙ ˙ e ¯ ν y L β β ˙ +2 | L x α pt ) ηE ˙ α β ? κ κ 4 ) ¯ ˙ ,... κ E n pt L α ( + ( 2 . , − ˙ E ¯ L κ β  ( ¯ ˙ u e  2 β + Φ ¯ , y T 1 ¯ L α ), we have y αβ r ηE 1 n ) . We also define the generating α ˙ X 2 α 4 κ  bh − E y  + 1 − r − iy + ( = ) := Φ +¯ 1 e η η ?L u 4.84 n = , it follows that L αβ n + 1 − ˜  = Φ L + 2 1 + κ x, y e := := ) αβ P ˙ η η β ? ( β ) L p ˙ y ¯ β ? y ) (  L pt pt α πiη ˙ dη 1 α β 1 b y L ) Φ = 2 ˙ κ Φ − α (  − ( E πi ˙ ) ) β 2 η ( π dη  = 2 L L L ( α T αβ − ) ˜ C ) κ L e  ≡ (¯ ( I L ,C , ) = – 42 – ˙ ? -independent, and defined α C = n κ L b y ) n ¯ L y Z I ( ν ( e L , =0 P  η (  1 2 , v ¯ π y is 2 β 2 | ? κ b ?L Φ ) 1+ 1+ + ) αβ E bh ? := − L − L ) + ( n β and ( ηE n n 1 u E y P 4 ( b 2 Φ 1 − α n 1) 1) ) − ) ˜ − αβ D ν L L ?L ) − E − r − ( ( ? e κ L ) := Φ ,... n = u ( 2 1 ,... κ = ) P -gauge and Kruskal-like gauge 2 (  − b + L , = 2( ), and using x, y ? α (  1 L ), whose spin-2 component is the Schwarzschild black hole Weyl L , ( L αβ β X y y , one has 1 1 b  ) Φ only consists of a rotationally-invariant dynamical scalar field. X r Using the regular presentation ( κ bh  − ≡ 1 = E η − ( 4.19 x, y n ? κ = pt C L , we can write  ( L u n L n 1 2 α L αβ bh 2 ) ≡ ηE := P ) 4  κ E C + ( L is a Killing vector with Killing two-form ( L n − bh u e 2 ˙ κ ) and ( P β Φ ( L α exp ) of v := 3.1 α p ], while Φ × ) 3 E αβ , = − ( ) 1 E , u ( bh ] for further details. Performing the star product α 1 ) D Φ E + ( u where see [ By introducing( an adapted spin-frame consisting of the where The matrix radial AdS coordinate tensor [ Black-hole sector. for the dynamical scalar fieldin and the the black (self-dual hole part andare of particle activated the) sectors, spin in respectively. As we shall demonstrate next, all spins recalling that functions 5.2.1 Weyl zero-form From eqs. ( where we have use the fact that 5.2 Master fields in JHEP10(2017)130 ) n . into  (5.35) (5.37) (5.38) (5.32) (5.34) (5.41) (5.39) (5.40) (5.33) ˙ β L α ) . 0 β σ y , ( 1 − αβ iη ) , αβ , L ) ˙ α − ) κ ¯ , y E ˙ ( β L ( ˙ α 2 α ¯ α y α D y x 0 Schwarzschild black x 2 r M 1 η   2 4 α η e iησ 2 , iy = 2 + x e ˙ x β 1 n − 0 AdS α − 1 − αβ µ A ) L − ) ) can be found to be , 1 0 1) 1 y L ) 1 iηx ( σ sector is thus given, up to an √ s − 2 2 κ det n (2 ( )( s δ ( 5.24 = s α  − ˙ = ) n 2 β ) 1 1+ α x, η y  E ( η − ( 1 1) 2 , u ), we start from the regular presenta- ? κ ) − β if + 1 − := ) ( E y 1 + ( e y η η 1 2 − with (5.36) (  u 5.23 ,B − αβ ˙ 2  ( ) of the delta function in non-commutative β ) = X δ  α  L n ) to the spin- β x in ( n πi µ A κ , B.1 ) dη α E 1 ( 2 +1 π  ) ) , f ( s pt 2 α − , – 43 – ) 1 ¯ x y r n ε 2 n y det )¯ 0 x, η ( 2 i 2 P x ( + 1 − η σ x C 1 2 η n 1 ( 2 = 2 η η I ν η f ∼ x,η η y iη ( + ) ε  = 1, this corresponds to an s + 2 = 2 0 + B − exp 1+ x 0 ˙ (2 ? κ n β β 0 + 2 − β α
πiη dη ) − iηx y n δ 2 L ,n,α ) 2 iηx L B 1 iηx  ¯ y 2 1) β κ 2 bh 0 1 1 − 2 (1) η − is real and imaginary, respectively in the case of scalar (odd x,η − C − α ( ( x 1 − C ( n 1 π iησ A I 1 − 1 A p µ 1 ( 1 1 − 2 − := := = − = √ n δ L n ˙ i 1 β ) = 4 i y In order to compute Φ A ] for the case f = α ) singleton twisted projectors. The Weyl tensors are thus of generalized =0 E 3 ¯ =1 y ) = ∞ β ], and the corresponding asymptotic charges are electric and magnetic, ( 2 n | M X n L 1 α det L y n ¯ y r 2 πδ e A P 0 2 , the contribution from ? κ n L iησ ) = ) of the twisted projector, which transforms under the adjoint action by − ηE -dependent real factor, by 4 L s x, y − y ( 4.85 e ( 2 bh δ C - and twistor space, the star product in the second equation in ( where Recalling the complex analyticity property ( where, in stereographic coordinates, As first noted inhole [ Weyl tensor together with its generalizationParticle to sector. all integer spins. tion ( where we note that and spinor (even Petrov type D [ respectively, for scalar and spinor singletons in the A model, and vice versa in the B model. For a given n we have Since JHEP10(2017)130 ) , ). ). .   2 2 5.26 / 4.23 4.18 (5.47) (5.44) (5.45) (5.46) (5.43) (5.42) it r 1 2 ) − 2 it e r − 1 + e ∗ 2 e ν . . (1 + 2 + after eq. ( ∗ 1   x ˙ β 2 e ν 2 2 y r )¯ , with the notable η x it 21 ˙ ), and using ( β η + 2 2 η | y e )¯ x + η 2 ( η from the ground state 1 + / ˙ | 0 β 2 1 4.83 x + , α 2 ( ) x ˙ and 0) can be obtained from ) ˙ β 0 e ν 2 β 2 =2 , M y ) | iηx it r α  )¯ a ,α η α 2 L n e 2 η | pt ( x | M pt iηx iy − φ x + α b − e 2 ( V ( ˙ = 8 0 β iy 1 (1 + D − α e + # 1 n 2 1 M ) iηx e , ν ,α  α L 2  ˙ is given by the coefficient of the 1 β (  1 bh iy 2 y b − e V )¯ x − + 1 + 1 − ( 1 =1 vanish and only scalar modes appear. i | η η = 4 2 + η η 2 n 1 | pt x, η x 0  (  φ − πi ˙ β 2 + 1 − dη − ix ,... 2 α α x 3 πi η η 1 z dη , 2 1) M 2 + 2 ) − = , x πi  ( 0 1 + 2 – 44 – dη ( + 2 C C − ix  α I , which are abelian; see Footnote I ?L y 2 1 ˙ 1 (1) β n − 0 ¯ α y − C 0) accompanied by its negative-energy counterpart in 1, then we recover the mode function of the ground e ν ˙ e ν -gauge b , e ν β S I :=  1 + 2 b  Φ implies that each positive-energy particle mode must 2 L 1 α ? (1 + 1 α 1+ − ) e ν e y 1 − 2 0 6= D −   e ν − σ n  ) ( n L 2 2 + i 1) ∀ x x 2 encode the rotationally-invariant massless scalar modes of 2 = ∓ − ( x 2 + − ) α > x L y 0 ( 6=0 2 α + | transformation to the internal connection ( X n b x S − ix 0 = n | ) 2 L 1 1 2 − ix = 2(1 = 0 for x  2 − 1 ) becomes = n 1 − =1 − e projectors, i.e. ν ,η |  n 1 2 | pt =0 5.43 2 1 − x Φ e ν e ν | P " α  = 2(1 . , i.e., all Weyl tensors of spin 1 e y ), we find ˙ n α pt y 0) together with its negative energy counterpart in = 8  and = 4 0), as ( Φ , , 2 5.35 1 (2 =2 =1 P | | and ¯ n − D n | pt | pt ( α φ φ 5.2.2 Internal connectionApplying in an adjoint and ( The projectors with energy where the last expression isnote given that in it the is regular globalof everywhere. spherically symmetric Similarly, the coordinates, scalar andthe mode we The corresponding realunity, i.e. physical scalar mode be accompanied byFor a example, corresponding if negative-energystate anti-particle of the mode; lowest-weight space seeD eq. ( We note that they expansion in oscillatorsMoreover, of the this reality condition function on only contains equal powers of which generate a (noncommutative)exception Weyl algebra for generic Thus, we arrive at and we have introduced JHEP10(2017)130 , T αβ D (5.50) (5.51) (5.53) (5.48) (5.49) (5.52) ) with defined is real), Lαβ 4 , 2 κ 5.30 1 r , that have − αβ 2 ) ) − G )–( . k AdS k , 1) ( n ( β n b 2 , z ν = e ν α − w k b k  1) 5.29 2 can be computed t 1 1 2 ( )  (   +1 − t − t 2 + L α b 2 b 2 i ) ) t ρ : one way of proving , L n,α αβ κ e α ( L − θ − z z α 2 κ − κ ) α w ?V ρ 1 L α ? z   +1 −   L β n + ( t t κ ] to be real-analytic in ) +2 ) 2 2 i 2 P D α L / 1]. The quantity k / k e αβ z β ¯ , αβ 1 y 1 α + ( α D z 0 1 2 ρ  D  β ) iz − z 1) αβ 2 [ L +2 α iησ =1 =1 ∞ ∞ z ? D κ Lαβ k − X k X ∈ αβ ( 1) − t L κ n 2 n D t ( − β L t Lαβ   1)  ηE ( z y 1 4 ( 1 κ α ( − z + 1) − 2 1) + 1 − + 1 − 2 e 1 t δ 1) i +1) ), after which + 1 t t ( was shown in [ η η 1 η η ( − 1 t − 2( t i − ( −  2  ) C.4 ( k + k ,α t + − L  ( ˙ ( )) )) i β 2 α , bh 1)! 1)! – 45 – πi 6= 0 and real (recall that ( 2 i +1) πi 2 L β 1 b dη t dη V 2 -dependent eigenspinors of v 2 − − β /t 2( − /t αβ − β x ) ) ) 1 2 αβ ε ε + 1) as in ( will be spelled out next. k D k L ( ( − α 2 in the spherical global coordinates of ( ( t ) D ? e C , and C κ 1 C L ) ( I I π/ n,α + β κ  + 1) β (log(1 − + 1 (log(1 ( ˙ β ε Lαβ V u ε ˙ t L α α t t 2 L = α 2 2 ( 2 with α ˙ ¯ κ v L v α i 1+ − ( 1+ ˙ ˙ ¯ θ v β α u − ¯ − 2 y = − y + αβ dt n a ˙ dt n G β + 1) ), is indeed real and proportional to cos + 1) ) 1 y ) G ˙ L α + ¯ t t − α αβ − αβ ¯ κ − ( − := 2 ( − ) +2 α (  ( G ˙ to write β ˙ 1 , u 1 L y β 1 1 ˙ L α = ¯ y The master field − n αβ − n + α  κ α αβ ¯ ˙ κ α X X Z i ˙ 1 Z u ρ β ¯ D y G i i ¯ y − αβ 1 2 × ˙ × 2 1 α := ( := G − y e = 4 = 4 +¯ α 2 := b αβ ) ) L αβ ,α ,α 2 G L L G κ ( ( pt 2 bh β G ) b b y V V 1 L α y κ  ), where singularities appear on a plane in twistor space. This can be seen by 1 are moved to the zeros of ( ( 1 2 − − p e = A.19 = t at imaginary parts provided that which push them awaywhich from is the given by integration the domain contractionthe of rigid the spin-frame ( and we have used The crux of the matter is that, after the star product with the projectors, the singularities where we recall that except at the equatorial plane in ( introducing a source using the lemma whose real-analyticity properties in Black-hole sector. and the particle sector by where black hole sector is given by JHEP10(2017)130 ) − L ; ) is y 5.53 2 θ ( ) as a 2 x (5.55) (5.56) (5.57) (5.54) z δ 2 ( 2 η cos , δ 1)! r ) + , ), thereby − k 2 e ( y n Taking into 0 − α − e − ν + 25 αβ e y k , u D 26 ( 1 = iηx β + α +1 − 2 ρ k t t u , i α  αβ ρ e − ), and consider  b 2 ) in the decomposition 1 D 1) (1 − − − i C.4 in representing k / 2 5.56 t ) Lαβ , given by a pole in the z  1). = +1, unlike in the case 2 ∂/∂z 2( κ 2 w Y   x t 1 -integral has been moved − t + , we need to compute the 2 t = t ) α / k 0 in that limit, and the Gaussian − ,α ( e y -gauge). To exhibit this, we L as in ( ) is thus given by 1 /  ( pt log L α → ⇒  α ) b 1, to V β ,α  1 ρ L D ( + 1) − , which yields the source term − 2 pt =1 β ∞ ? t b k X κ ρ ] V i 1) = ) 1 ) by realizing the delta function as = +1 in the exponential, it follows n t )+ 1 dt + β − ), and (1 − t -integral is convergent away from the t L  u ( t t − ? κ , ∂ 5.53 1 = 1). ( + α b ) S α ) . Thus u , α ˙ z 1 k 5.42 + 1)) = ( β 1 + 1 − )–( L ( z We remark that the singularities are can- ( pt i − α − t iφ α ) ) η η ( − Z ρ Φ −  L β i ) ( 5.50 e  M b e y − +2 2 b + S ) over the rigid spin-frame ( D z − 1 ( αβ α αβ πi i −  – 46 – ) = dη D e − − β y D 2 , which can be cast in the form of the first factor in ( E α t β u = − − ( − ) ]. Thus, the e − y ( z ε e ˙ y − − α / 1 1 β ( α z , u + α +1 − z u C )(1) e t t y + v α ). The final form of i  1) L ) I e i y iφ 2 , ( + 1) ( ˙ pt e − − β E i are defined in ( e ε t t ˙ + ( ( e b α e V ( ( 2 )  1 θ 1 u α ( 5.45 1+ +  − e y 0 e + y M ∂ − + 1 − 0 → i +1) n D 2  t t t → + T  x 2( 1) + sin 2 2 . Interestingly, the singularity in the M ) η − , while it has singularities on a plane in x lim ( ) − β ( 2 ,α i  2 ? e L Z u A )(1)+ + η ( ,... pt x − 1 α L 2 0 ) b + ( ( 2 ). To this end, we introduce a source V det pt +  = L To analyze the singularity structure of − x α b , V 0 ¯ ˙ y 1 ) can be obtained directly from ( β θ u ), which is regular (and exact already at the first order, as the Weyl  1 iηx X + 0 ˙ − α  ] and appendix E in [ ) = u 2 5.49 ∂ κ e y 2 = 1 iηx ) reduces to lim ( 5.55 ? κ , ¯ n − 2 iησ 2 := i ) δ 2 cos αβ 1 regular in 2 L h − − A 5.53 ( D  1 r ∓ i ) is  1 × L b Φ det L y b ) ( = = π = ( pt ,α 1 L 2 b 2 ( V − pt  δ ) = αβ ) b V L = κ L Lαβ ¯ (1 y The formula ( The solutions contain two sources of explicit Lorentz symmetry breaking: one due to the expansion ( i pt 0 κ 26 25 b 2 V over the generalized projectors,delta and sequence; another of one thesewith due two, respect the to to latter the the is introduction spin responsible of frame. for the differentiησ signs in ( with determinant ( consider the linearized equation first order of the perturbative expansion (i.e.Singularity-free for twistor space curvature. celled, however, in− the star commutatorzero-form [ does not receive any non-linear corrections in the where from its position in theof holomorphic the gauge, black namely hole at sectoraccount where it the was more removed except detailedthat at structure the of equatorial plane. the poles at where the modified oscillators regular, as shown explicitly in ( Particle sector. star product in ( obtaining for details, see [ equatorial plane, and one is led to the conclusion stated above. this is expanding the spinors ( JHEP10(2017)130 ) .  5.26 =+1 given 1 (5.64) (5.61) (5.58) (5.62) (5.63) (5.59) (5.60) η

−  . =+1 − = I e η y η ?κ

! + ) 1 − e y ! + e ) y − 1 1 z k + +1 − = ( − n t t − e y η , i e ν y =

1 e η − k +1 − −

t t ! 2 e y − ˙ i β  = 0, and that, as 1 z ), and using ( ¯ + y e 1) b ) at first order. 2 , − ˙ e y + β dt 2 − ∓ e ) thus reads y 1 , = e ˜ α L, L z t ( − e y η +1 − i ) ( w t t 1)

4.83 M ? e ) 2.67 i 1 1 ,α α 1 − b L ) dt e +1 − 5.57 − e 2 y ( − ,α   t t iy pt L 2 x i t ( + Z 2 e b pt V ( e y e ) / k 1) + b V 1 2 , 1 + 0 1 ˜ 1 dt L α 1 + − +1 − x z − ? i t t t −  ix e 1 ν ( − i iz ) 2 + 1 ,α Z e y 1 e b 0 2 L 1 − 1 ) ( =1 − − ∞ bh 2 = − ? ix e e ν k X + 1+2 L b − V Z z z L ( n 1) − ) + − i has a pole at e = y + e y  dt 1+2 + e 2 y ηE + − = 1, i.e. ) ( z − 1 4 e y i  ,α t b | L + − − 1 − e − =+1 ( ( e y pt z n ) to compute + 1 − +1 − η e 2 | β t t b 1

)(1) V − + 1 x i z ) η η 1 e y L − − − =+1 e β ( + ( + ), respectively. Thus, more explicitly, − z pt e Z y η i  z 5.44 α 0 k 2

b + e V + ) ˙ − ˜ e β y L e y ix 2 )) e y + ¯ 1) y ( 1)! 2 1 – 47 – πi 2 i ˙ z x 5.49 β − dη e ν . The left-hand side of ( = 2 dt e − e − α − L, ˙ − /t − β e y 2 ) 1 + z t b M L ? ε y  x − k ( − ( + 0 α  e y ( e e ? y y − ) + 1)¯ C 1 ,α ) ( 1 iy z 1 ix 0 + i L 0 2 I α − e ( e  ν e + ) and ( 2 y e bh ), we use ( b (log(1 x ix e S y Z 1 ε b 2 V ( 2 2 2 − − = i ? − x − x e 1+ ? ν e e y 1 1 5.48 + 1+2 5.57 + 2 (1 1 − 0 − + 0 1 b  dt n vanishes except for x 1 − + 1) x, η e e ν y b ) e L + ν ix ) ( ix e L t n 2 ˙ 2 + 2  β − ( e = 4 ν , we find = x ( − e y 0 α − = 1 e 1 1 L ) − 1 transformation to the internal connection ( 1 − n − b ix L M ) ,α − (

X α Z  b L b L e given in ( ν )(1) ) ) b ( i S (1 + bh L? 2 2 L × ) b ( V ib α ,α x x pt L = 1 + 2 y 2 ( b = 4 V pt − − b b  − L V + ) ) is satisfied provided that = ˜ ,α ∂ b L 1 ( = bh +  b V  = 4(1 = 4(1 and 5.57 = η ) and | ) ,α )(1) ?κ ). This means, in particular, that )(1)+ L α ( L e y L bh ( ( pt =1 b pt 5.35 V | b b V V n 3.35 | pt + Φ ∂ where the black hole and particle contributions with 5.2.3 Internal connectionApplying in an Kruskal-like adjoint gauge and ( which holds onin the ( grounds ofanticipated, the the latter first is of exactly the cancelled by properties the commutator of in the ( distribution Thus, eq. ( As for the right-hand side of ( where in the special case where JHEP10(2017)130 ˜ L ] for 1 aligns (5.67) (5.65) (5.66) e L . is related ) αβ ). However, ). b L ) , D ( k pt X ) contains the ( β n ˜ L z b V ρ e ν , w 5.67 α while it has sin- k 1 ρ is concerned, one of +1 5.53 − αβ  t t ) 1) Z i b Z 2 C − E i e 2 ( t − ˜ L α in D 2( β ) ], and then show that it + z -gauge, the star product ? z L = 0 are gauge artifacts. 1 α   ( α e ) y pt . L 2 r ) ) (see appendix E in [ z b ) L α V / k α b ¯ 1 2 L y 1 β ( 0 bh ˜ L  b 5.30 V β := γ -integral. To study the latter, we iησ =1 ), thereby modiying the quantities ∞ ˜ D t L z k X γ , the imaginary part of which only − ρ n 2 ) transformation induced by 1 L 5.66  is real-analytic in 1 − C 1) y . t 1  ( , ) ˜ L ,α Z 2 α − b L − z + 1 − , w δ ( α pt t α z ) η η in b 1 ( ( α ρ V i E 2 , as first found in [ −  ( )  ) ) − k +2 ), as in ( L u β L L ( (that is a modification of the singular plane in α ( pt e y )) bh ) αβ 1)! κ – 48 – b πi ) 2 = V b V dη E αβ 2 X E − ( ) ( ], the SL(2 α are collinear or not (point-wise over /t − ) E 1 + ( D β in ε ( , u k ( requires the detailed calculation in ( β ˜ Y ( L D α z C viz. ) α β Z αβ α Y I ) e y -measure out of the integration domain, independently E z  ), + ( t 1 (log(1 E ε 1) ( and +1 − u 2 2 t t − 1+ D t i 2 ( Z 5.55 − e  dt n + 1) 1 ) Lαβ , u t i +1) depending on κ β +1 − ), we conclude that t − ( t t ( z 2( ). In what follows, we shall first recall how the conjugation by 1 1) Y 1 β β 2 − n As observed in [ α α 5.55 x X − Z ˜ ? e -gauge at the equatorial plane away from L 2 ˜ i , that brings the solution spaces from the holomorphic gauge to the L 2 As explained before, the singularity structure in with that of ( η L ) × t b ). The resulting modification of the determinant in ( H ( 2 L := + i Z ¯ = 4 y x 0 ˜ L 0 α ) − − z 5.52 ,α L? b L 1 = 0, which is thus the only singular point of 2 ( iηx pt iησ 2 b = r V − = 0 is the only singular point of the Weyl zero-form, we conclude that the − b L G + 1) 1 r y ) does not generate any imaginary part, or any other sort of contribution that ) and ( t ( π 1 2 2 δ We remark that as far as the real-analyticity property of As 5.55 5.51 = -gauge). the details of the singular plane in 5.3 Vasiliev gauge Let us investigate thefunction mechanism whereby the (large) gauge transformation, with gauge may argue as follows:in ( unlike in thepushes the black singular hole points sector,of of in whether the the the spin-frames in Thus, comparing to ( gularities on a plane in L Particle sector. to the nonintegrable divergencies ofneed the the measure modified of version the of ( factor ( vanishes at singularities in the preserves the real-analyticity property of Black-hole sector. the spin-frame in in ( with where these equalities follow fromthe the explicit definition form made of inremoves unphysical ( singularities in and JHEP10(2017)130 (5.76) (5.70) (5.72) (5.73) (5.74) (5.75) (5.68) (5.69) (5.71) , 1 − and Vasiliev = . η ) | L ), we get )(1)+ , ¯ y L ], and shown to be a ( 0 , ) , + idem ¯ b y V 91 iσ α 0 ). − z z α − iσ , =+1 η α ity 3.29 − . +

tz z e α  − −  tz 1 ( ( =0 ity . + h.c. = 0 2 , 2 r = 0, one finds e )(1) − δ | + ) ) ) 1 L e )+ (1) u ( z b ¯ ¯ ν i y y G b G dt δ ( b e + h.c. b − V α ( H 1 e  b y α V + b tz, tz, 0 V ) = ∂ α z Z )(1) − − − − + ( ¯ y ( ( Z L z z + E ( − − α 4 − bh bh ∂ z b tz, − V − e y − + Φ Φ α e )(1) − − e z y e z u ) L + α )( ( b ) α β G 1 y 6= 1, for concreteness, it reads + – 49 – ( = dt t dt t − ¯ b y ∂ V 1 z 0 1 1 b β V n + -integral was analyzed in [ α = 0 the internal connection is instead singular, as 0 0 σ ? κ t + z = z ∂ ∀ + ( Z Z e y 1 α r z + 1  α α e y z P − z z iν )(1) 0, and to reproduce ( − e  y 1 b G = 0, 1 − := ( ∂ ν = = + α = n e y ]. At b e u V = r > ν − = =0 2 (1) )(1) r  ,α | + h.c. = x 84 b =0 ) b G in the internal connection and restores the manifest Lorentz , =0 for H r ( ) L r bh | 8 2 ( + b | G b )(1) C V ) ( T x At linearized level, the gauge transformation reads ,α 0 b α ) and decomposing using the spin-frame, yields b V ¯ G y )(1) b ( V ,α bh − ix b G α = b V ( 2 tz, bh 1 z 5.69 b  V − − and using that, by definition, ( 1 α )(1) bh z b 1 G into ( ( e ν 4 b V ib (1) − b H = (1) pt Turning to the linearized particle sector, upon defining b H we compute which has singularites on a plane in twistor space. and, inserting Φ that is, the linearizedgauge, internal recalling connection that obtainedgauges; the by see, linearized direct for integration example, Weylexpected. [ in zero-form Examining the is the Vasiliev case the same in the In the black holereal-analytic sector, function the on resulting Inserting Linearized analysis. Contracting by removes singularities in covariance, broken by the introduction of the delta sequence ( Vasiliev gauge, defined by the condition JHEP10(2017)130 , C can (5.77) (5.78) (5.79) (1) b H ; in G , Y 1 : − 1 = − X → η = ]. | : η ), as required by the | 95 L , T ) T . Two related issues are So far, we have set up a + idem Ω( | W + idem  µ ; ) 1  ˜ ν b G 1 ( ν, − e b u f i − e e u u 1 i i e u − e i ) e 0 ). It follows that −→ − S ( − e u ) i , 0 e u e 4.14 i α T  e z  α Ω(  | e u . Moreover, in the Vasiliev gauge the manifest z α µ ity e u ; ) ), can be mapped to Ω( z . The appearance of a pole in C ˜ e ν 0 b 2 G – 50 – ( corrresponding to black hole modes and massless ) Y 2 ν, x T b ¯ y f on x ν 2 + raises the issue of whether the master fields in the µ 2 + x 0 tz, H x 0 S Y −→ − − and ˜ ix large Kontsevich gauge transformations ) − 2 ix ˜ 1 ν ν L in 2 Φ( 1 ( ν, − in order to ensure the existence of invariants based on b f − 1 is large, that is, affects the values of invariants, and whether (1) pt 2 L dt t 1 everywhere b 1 S b 1 H G ◦ e ν 2 1 0 0 b −→ e ν 2 Z S b − α ˜ ν − z but has a pole in 0 ν, = b f = = Z  ) of exact solutions in holomorphic gauge, built from generalized Fock b G ( and traces over the extended Weyl algebra )(1) ˜ -function in the particle sector ( α ν b b δ 0 G V ν, ( b B f b V denote Kontsevich gauge transformations, which act on horizontal sections of implements the gauge transformation with gauge function denotes the map from Weyl to normal order; -gauge, which may be related to those recently proposed by Vasiliev in order to b S L 0 L the families of master fields making up the exact solution spaces, we would like to S S b f Let us outline the pending steps in somewhat more detail. We thus start from the The singular nature of where particle modes, respectively, whichLetting can be givenand equally well in Weylestablish and the normal following order. sequence of in the obtain a quasi-local perturbation theory in terms of Fronsdalinitial fields family [ space operators using parameters Vasiliev gauge, which are elementsassumptions in made Ω( inintegrals section over whether the gauge function it induces redefinitions of the inital data for the Weyl zero-form and spacetime one-form first discuss the prospects of extendingwe the will solutions to propose the a Vasiliev gaugeorder correspondence to on all to order. the an Then quasi-local alternativereproducing branch perturbative the of scheme deformed based the Fronsdal on theory theory, and on-shell. normal that these two equivalent schemes are in agreement with direct integration in theVasiliev Vasiliev gauge, gauge just beyond asperturbative in the the scheme linearized black that hole approximation. provides case. solutions in Vasiliev gauge up to first order. We will which is indeed real-analytic Lorentz covariance is restored, and we can write be traced back toconnection the is fact a that the curvature deformation Ψ that builds up the internal which is regular in JHEP10(2017)130 b µ f ), )– H T (5.80) 3.15 valued in ), where b B f to the Weyl ( b L (2) W b . Φ ) Tr ˜ ν B 0 ν, b R f )( L ] = S b f [ , but more generally it may can be transferred from ◦ S 0 Z µ S ) must be an element of Ω( M ◦ , which is thus a gauge function )(2) µ µ b G H ( creates the asymptotic anti-de Sitter S is thus a set of forms on b Φ ( ◦ ) L 1 L?M 1 T − − ) Ω( ) 0 | , the resulting component fields should be 0 := µ S ; ) S ˜ ν µ b ( W G ( ν, L , where b – 51 – f up to a homogenous solution parametrized by a b b H L ◦ ( . µ and induces a second order correction G W H L? ], in this paper we have obtained a solution space to µ Tr (1) = 2 B , b H ?H 1 Z b X → G : 1 − µ ) ] = ) µ M T -dependent part of ( M Ω( | Z µ ; ) = ( ˜ ν b G restores Vasiliev’s gauge, in which the asymptotic field configurations -dependent piece cannot be corrected using any second-order initial ( ν, µ b f Z b ˇ [ H H ). The simplest realization of this condition would be a Weyl zero-form S Z ( 1 L . Thus, the 0 implements the gauge transformation that bring the solution to Vasiliev’s gauge, µ H region, and which fixes the gaugelarge function gauge function S by defining ), based on separation of twistor space variables and auxiliary integral presentations The large gauge function Our solution method combines large gauge functions with the Ansatz in eqs. ( Extending the methods of [ As for the existence of the map, a nontrivial compatibility condition arises already i) L , for which we can compute an on-shell action as a functional of asymptotic data and 3.17 ( of intial data. More precisely, our construction involves: hole and scalar particle modes.that the Interestingly enough, scalar in modes thearises give gauges already rise we at use the to we secondpoint a can of of backreaction observe classical view in perturbation of the ordinary theory.non-locality gravitational form This of theories, effect, the of can vertices unusual black be extracted from holes interpreted from the as Vasiliev’s modes, equations a that at consequence every of perturbative the order. In what follows, wephysical first interpretations and summarize prospects our for results, future after research. whichVasiliev’s we four-dimensional turn higher spin to gravity omitted by superposing details, spherically-symmetric black it would also befix the interesting precise to formwould examine of provide to the an Vasiliev what gauge intrinsicthe extent condition method theory. the beyond for leading compatibility fixing order; may this if actually apparent so, ambiguity then currently this 6 plaguing Conclusions and outlook data for Φ that is, uponelements expanding of it in thethat approaches basis a of constant valueturn at out the to point be at infinity necessary of to allow integrable divergencies. Proceeding to higher orders, to including a boundary state. at the second orderzero-form, as whose follows: collectively denotes the master fields, we have We note that the moduli (boundary states) entering via If the map exists,W the final configuration invariant quantities. Thus, denoting the on-shell action by JHEP10(2017)130 0 b A )-rotation C , , while the deformed Z using a regular presentation that Y -gauge, reached from the holomorphic gauge L – 52 – presented via a generalized Laplace transformation, Z ], which will give rise to more general Type-D solution 1 and the spacetime gauge fields to higher orders, which requires b H is expanded over an algebra of projectors and twisted projectors in ) listed in [ 0 b C A (4; oscillators. In such gauges the Weyl zero-form is first-order exact, and the , which is regular on twistor space away from the origin for black hole modes, sp ). Z (1) alone, and in a Kruskal-like gauge, reached via a further local SL(2 b H L 4.88 an extension of Vasiliev’s gauge,branch and of the moduli study space oforigin in whether for there nontrivial which particle exists the modes. a spacetime quasi-local gauge fields remain smooth at the lator algebra, to which we see no obstacles. shell theorem. whose exterior derivative reproduces theture. inner Klein operator in the two-form curva- supersingleton spaces realized asimplements the functions normalization on of states in compact weight spaces. oscillator is a distribution on zero-form and a deformedalgebra twistor whose space enveloping connection,in algebra obeying ( is a a deformed subalgebra oscillator of the star product algebra interpretation of its content in termssubtleties of mentioned black-hole in and the particle Introduction, modes, barring more all transparent. the twistor space in a holomorphic gauge, where they consist of an undeformed Weyl tion and singular in thefield twistor configurations fiber to space all for ordersvia particle in modes. the We haveof also the studied the consist of unfolded (free) Fronsdal fields. We have determined the linearized contribu- The algebra In the holomorphic gauge, the Weyl zero-form is constant on Computation of the gauge fields to first order, and verification of the central on-mass- Prior to switching on the gauge function, the field configurations are localized to Computation of Extension as to include general particle and black hole modes in the deformed oscil- c) a) b) ii) iv) iii) More generally, we would likeexample to the apply generalized the projectors Ansatz built togenerators via more the of general different inequivalent initial choices data, ofspaces such the with Cartan as fluctuations. for The construction above remains to be completed in three main respects: JHEP10(2017)130 1); − , s 1 − s ] that reflection 87 , that is, by a twistor space Fourier y κ 3) into compact weights filling up the , ] that the Flato-Fronsdal theorem, which . Thus, the aforementioned association (2 3) irrep with highest weight ( Y 87 , symmetry between the modules embedded so (2 2 Z so – 53 – It was observed in [ ] suggests that the transformation to another theory that we propose to associate to the 117 2 , Z ). ¯ y contains the (complexified) 0 115 , s iσ 19 − y ]), i.e. a Hilbert space, while the black hole sector is real vector space with positive ]. Turning to the compact bases, indeed, the massless spectrum remains spanned ( 2 δ 107 114 Having obtained a classical moduli space consisting of particle and black hole modes, We would like to stress that the particle and black hole spaces are isomorphic as com- Concerning the interpretations of our results, and future directions, a number of re- ∼ , while the reference state of the adjoint representation is given by the twisted projector 1 1 e black holes (or, rather,with its to Euclidean the structure. black-hole microstates, as discussedit in would the be Introduction), desirablea to higher-spin provide gauge-invariant it functional with of a the free energy particle (or and on-shell black-hole action) deformation expressible pa- as bra [ definite bilinear form. Themal correspondence fields between star [ productinto algebras the and oscillator free algebra confor- hastheory a at holographic the counterpart, boundary wherebybe the of mapped free anti-de by conformal Sitter a field spacetime, with its Hilbert space structure, can are the realizations ofthis anti-particles, can while be the madein black more the hole manifest black states by hole formsector looking sector real is at and a vector the complex twisted-hermitian spaces; adjointrealize vector in Ψ space using the fields, with a particle a which suitable sector. positive are extension definite hermitian of Thus, sesquilinear the the form supertrace particle (which operation one to can the extended Weyl alge- plex vector spaces; indeed,product multiplication the with isomorphism the inner istransform. Klein implemented operator However, as by far means asis of the a real dichotomy, one-sided as structure it star as mustthat higher be spin particle imposed modules states using is the are concerned, twisted-adjoint realized reality there in condition. terms It of follows complex operators, whose complex conjugates see [ by basis vectors that areP real-analytic, with reference stateP given by the Gaussian element both of these representationscompact admit bases, two connected dual byelements means pairs of are of harmonic operators bases, expansion. that withrequires In are Lorentz that the covariant in polynomial former and the in basis,Lorentz free all spin theory basis limit, the generalized Type D sector of a Fronsdal field of adjoint representation of thestates higher to the spin black algebra. holeof sector. one We of To would this the like end, twotations spaces we to into begin in operators associate the by on the recalling direct whichthat from product the latter is, higher [ to one spin the should algebra dual think acts space, of in turns them the these as twisted two being adjoint represen- terms fashion, in a twisted-adjoint zero-form. Moreover, marks are in order: Black hole microstates. states that the massless particlesingletons, spectrum has is a contained natural inanti-supersingleton, the generalization which direct to decomposes product the of under direct two super- product of a supersingleton and an JHEP10(2017)130 ] 95 is inti- (1) b H is essentially ]). We recall (1) 18 bh , b -gauge directly in 1 H is, i.e., how far the L b H , which are indeed independent only form independent variables bh s M in the internal connection and hence Y – 54 – -gauge to the Fronsdal frame. At the level of the L -gauge and the Vasiliev gauge as two different frames L -gauge to Vasiliev gauge has no interpretation in terms of from the twistor space connection, while the master fields , i.e. the eigenvalues of Ψ and multi-black hole solutions (for which we have reason to L n We have obtained different solution spaces to classical equa- Y ν bh In terms of the gauge field equations, the role of -space, and interpret the equations of motion in fields from the . Y induce a redefinition of the initial data for the zero-form and for the s c H b H ] for recent holographic tests of the local second-order interaction terms 97 , ]). We plan to assess how large the gauge transformation = 0, since there are inverse powers of 96 95 ], which are known to contain higher spin amplitudes in their leading order. Y -function source in twistor space. It is expected that, extending Vasiliev’s gauge δ 106 [ inherits a singularity in The identification of the asymptotic charges requires the large transformation connect- In the particle sector, on the other hand, one simply cannot expand the master fields Z (1) - and Vasiliev gauge are from each other, by examining its effect on observables. As a b first step, it isin natural to examine zero-form charges, that is,Why the decorated Vasiliev open gauge? Wilsontion lines of motion intion different space may gauges have that its are own set presumably of far classical from observables each (finite higher other. spin invariants); Each for solu- spacetime one-form, to berewrite compared the with non-linearities those recently encoded(see proposed in also by the [ Vasiliev equations infound in order in to terms [ ofL current interactions [ H in Vasiliev’s gauge aremately completely related regular. to thehas The holomorphic a origin gauge we of start theto from, singularity higher in orders, in which the two-form curvature terms of the correspondingset enlarged of set Fronsdal of fields as component a fields, subset. thus containing theing original the holomorphic gaugethese to transformations the in this standard paper. Vasiliev gauge. Interestingly, in the At particle first sector, order, the gauge we function have found around in the generating functionlinearized for transformation the from gauge fields.the original In Fronsdal field other content. words, Instead,class it in of would the functions be on interesting particle to sector introduce an the enlarged to rotate the spin- Fronsdal fields, this amounts toof a different generalized ranks Weyl andsector, rescaling gradients that we of mixes can the Fronsdal interpret physicalfor tensors both scalar. Fronsdal the fields, of In which other the words, Vasiliev in frame is the preferred black as hole it is diagonalized. is in terms of thevariables. parameters On the other hand,in the the asymptotic free charges theory, i.e.individual (linearized) the Weyl physical tensors characterizations is of reliable black onlyLarge hole asymptotically. nature microstates of in terms of as holographic correlation functions obtainedthermodynamical relations from for multi-particle black states, holes andoff-diagonal from various operators derivation ensembles of into constructed Φ by including expect that there existthat, a as “dilute noted gas” above, approximation the at gauge large invariant distances characterization [ of single black hole microstates rameters. From it, a number of physically interesting quantities can be obtained, such JHEP10(2017)130 (2) b Φ ), but it Z ( 1 L -dependent piece of Z The boundary value formu- In a similar fashion, starting from – 55 – . Z -gauge and other gauges that one may surely think of as well, they may L -gauge, for instance), in the sense discussed above. Moreover, they may admit L As for the Comparison with quasi-local Fronsdalour branch. solutions, it would beto very eat interesting to up seethe whether the one dual black could boundary use hole thenon-linear conditions modes gauge field in freedom at configurations Weyl every expressed and order only normal in in order, perturbation terms thereby theory, of being while Fronsdal left imposing fields, with that fully could be order and expanded overmanifold. the assumed This class is of aat fiber form leading of functions order, litmus and which test, means zero-formsin since that on the there there Vasiliev the gauge. is is base no not Uponneed ambiguity ambiguity going not in in to be the Weyl the real order, Vasiliev analytic the gauge on latter should belong to Dual boundary conditions in Weyllation and that normal we order. are tryingconfigurations to with implement finite involves star dual product boundaryones and conditions, traces that connecting defined are field in real Weyl orderwhether analytic with the at corresponding second the order origin correction of to twistor the space Weyl in zero-form normal can order. be A mapped key back check to is Weyl spaces their physical interpretations.interpretation may Finally, instead as be forWe shall in the comment terms holomorphic on gauge, of generalizedof various its correspondence Vasiliev’s deformations space physical equations geometries towards of the and symplectic end. related manifolds. extensions the topology of asymptoticallylike anti-de gauge), Sitter and/or spacetimefor whose (as the component is field thetheir descriptions case proper go for set beyond the ofexist Fronsdal Kruskal- classical suitable theory action observables. (as principle It and is related reasonable free to energy expect functionals, lending that these there solution may also this version of the theory thatfuzzballs, we as envisage a discussed possible in interpretation ofin the them a Introduction. as future gravitational We publication. hope to make this proposaldescribe more field concrete configurations on correspondence space geometries that are no longer having form of a conformalto field the theory boundary that conditionsconformal of has the theory been bulk in deformed theory. these bysymmetries, Thus, sources operators if then is with the the an perturbative sources sourcesthe expansion expansion dual may existence of around correspond of the a such to freerequired a Fronsdal theory for free fields with Fronsdal energy fields in higher functional asymptotically. the spin would As bulk. single for out the Therefore, the higher Vasiliev spin gauge, black as holes, it it is is within observables that involve spacetime gauge fields.point Thinking of of a the path classical integral theoryon-shell, adds as which one a more can saddle condition, then namely beclassical the interpreted observables finiteness in as of a question. a classical freewe If action energy have the reason given quantum as to theory a expect in function on question of physical is the grounds gravitational, various that then there exists a holographic dual in the example, the holomorphic gauge supports zero-form charges, but gives trivial results for JHEP10(2017)130 . ]. b J ] for 3 102 , 95 – 92 dependence. This Z Our method starts from and Y We have presented so far a ]. The main differences between the ] and relying on a different type of 94 , while they arise in the normal-ordered 102 b H The higher spin black holes are constructed on – 56 – ] that they are regularized by the homotopy integrals following the ]. 102 111 – 109 As we have seen, thereity is structure an on intricate the interplay base betweenfield the manifold. equations fiber containing This algebra a suggests and dynamical aspaces two-form the higher that singular- of spin can landscape be more formulated based generalnamical on on topology correspondence master two-forms and have with been more introduced general using noncommutative three-graded structure. internal Dy- Frobenius algebras or light-fronts. Generalized higher spin geometries. correspondence spaces with basefiber manifolds functions of given a by definite operatorslator) topology in including Fock the using spaces inner definite (the Klein classes Hilbert operators of space introduced of via the the (non-dynamical) harmonic two-form oscil- oscillators from the start — ison also related the to allowed the class problem of of functions,We determining gauge a expect proper transformations there restriction and field to redefinitionsdual be [ observables, including a the correspondence on-shell action, betweenfields but the not — two necessarily similarly at approaches the to only level what of at the happens the master evolving level classical of fields along either equal time slices procedure can be contrasted withVasiliev’s equations the in aforementioned the normal-ordered sector approachthe to latter), of which solving particle is and already large blackprecise enough hole relation to modes provide between a (see these physically for nontrivalaccount two comparison. example that approaches The [ the — gauge-function which method in introduces general other also non-polynomial needs functions to of take the into Comparison of Weyl- anda factorized normal-ordered internal approaches. Ansatz (whichtwistor is space) tantamount in to order startingmeaning to from that highlight total lie Weyl the at ordering the generalizedafter in heart projector taking of our algebras the construction, of star and special arrives product at physical explicitly, normal-ordered thereby quantities mixing the be regularized using the regularbeen prescription proposed following the in hybrid [ approach,normal-ordered whereas approach. it has We expectlook that different as the distributions two inlevel approaches twistor of lead space, amplitides, but to as that master they they fields rely may that on nonetheless different agree trace at the operations in twistor space. (super)trace operations for computingtwo invariants approachs [ are as follows:approach in a) obtaning Concerning perturvative homotopy expansion integrals,approach for these in arise obtaining in the thetreatment perturbative hybrid of expansion of star the products master of fields; operators b) in Concerning the the Fock and anti-Fock spaces, these need to holography [ Alternative perturbative schemehybrid in perturbative approach, normal combining Weyl order. operation and on normal the extended order Weyl algebra. andformulated relying However, there on on-shell also the exists entirely a trace perturbative within approach normal order [ more directly compared with the non-linear results on higher spin gravity derived from JHEP10(2017)130 0 0 = , 0 3 , (A.1) (A.2) P 2 , = 1 ], that in , , being the E ab λ 108 = 0 M 2 . Decomposing iλ 2 with λ A, B = 3 , a 0 ? 0 − ] b AB 3 as with , λM ,P M 2 a , AB P = = 1 [ , † 1140296, Conicyt grant DPI energy generator a M ) P o 4 = 0 AB , ] M a AdS ( a, b P b [ c 3), and , iη , (1 ] 3) generators 1) with D | , so = 2 ] − A ? ; ] c background – 57 – M ab η B 4 ,P [ ] a class of differential Poisson sigma models with | ab C 78 [ = ( M iη [ AdS AB η ] = 4 ] in which SO(2 , ] d CD 87 | ] a ,M M b [ | AB c [ M [ iη = 4 radius related to the cosmological constant via Λ = ? ] 4 generate the Lorentz subalgebra cd ab AdS ,M M ], leading to a more general class of Frobenius-Chern-Simons theories [ ab M 107 [ where inverse further under the maximal compact subalgebra, the We use the conventions ofobey [ which can be decomposed using work of P. S.20140115 is and UNAB supported internal by grant Fondecyt DI-1382-16/R. iRegular grant N A Spinor conventions and Raeymaekers, E. Sezgin, E.D.Yin Skvortsov, for stimulating C. discussions. Sleight,Grant The M. Agency research Taronna, of of M.A. Czech C. Republic,ence Vasiliev I. project and Foundation was number grant Y. supported P201-12-G028, 14-42-00047 by andMoscow, in the that by association funding C. the with I. from Russian wishes the Sci- to Lebedev thank Physical for Institute kind hospitality in at various stages of this work. The mechanics based on underlying topological field theories. Acknowledgments We are grateful to R. Aros, R. Bonezzi, N. Boulanger, D. De Filippi, V.E. Didenko, J. new models whose Weyl zero-formsmatrices lend of themselves various to physical quantumthat one interpretations mechanical may as propose systems density as obeyingrived natural from deformations nonlinear the of equtions the Schr¨odingerequation. von Itand of Neumann-Liouville would to equations motion be investigate de- interesting whether to there make exists this a relation more clearer general framework for nonlinear quantum extended supersymmetries generated by thespondence Kiling vectors space spanning has the been fibers constructedFrobenius-Chern-Simons. of and the The proposed corre- fermionic as zero-modes a ofand first the central quantized sigma top-form description model of on generatesnition the a the of closed fiber master space fields useful thattheory. for can We a would be globally like used defined to formulation to stress of that provide Frobenius-Chern-Simons in a the more landscape general of defi- model that we envisage we expect their turn can be embeddedalgebras. into a As broader for classfrom the of the models noncommutative based quantization structure on ofbeing of homotopy the differential associative two-dimensional Poisson base topological manifolds manifold,Poisson sigma that it manifolds. models, in arises In with their naturally particular, turn target in are spaces [ special instances of homotopy in [ JHEP10(2017)130 and (A.9) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) β (A.16) (A.17) (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) A αβ  , , , ˙ ˙ ˙ δ β β = 3. In terms of ˙ ¯ α α y , , ) e 2 α , ? ,  ab , ∧ A ˙ α β  ˙ σ ˙ β ˙ ˙ δ y α β ˙ ˙ ˙ . . α β ˙ β = 1 α ¯ ) ˙ ˙ R β α β ω , and van der Waerden + (¯ ¯ ω ˙ , ˙ ˙ ) β α α ˙ ab ˙ , ˙ β ˙ a β ) α  α + ¯ , ˙ αγ r, s σ α  ) δ σ ) (¯ ˙ ) ˙ ab  β ( β i cb = ˙ ab ab γ ¯ β σ C y ab 4 λ ω † η − σ ¯ e y σ ) ? with ∧ ? . AB ∧ ˙ = = = (¯ = α , , αβ ˙ + (¯ c ˙ α ¯ α y ˙ ˙ + (¯ † a  β β ˙ rs a β ¯ α α y ) αγ ( ˙ ˙ α α β α ω αβ +(Γ ˙ ) ω ) ˙ M R β αβ e α αβ b ˙ ˙ ˙ α α ?Y R ) cd α + ω ¯ αγ σ +  R α ¯ α ω σ ( α a ab ) ˙ ) C 1 2 αβ (¯ β ab αβ , 1 2 α a ,P a ) σ P ) α Y ˙ α β γ σ a σ + i ) AB ab (( dω ( δ ab + ( ˙ de abcd a ˙ β 1 η R αβ σ 1 β σ ˙  β ¯ ) y ( σ ¯ ( − y = − ¯ y 1 2 := = (¯ = +   ? λ λ ? ˙ ? ), their realization is taken to be β AB ˙ 1 2 αγ 1 2 α ab − − ab α α βγ  α ¯ y y – 58 – ) (Γ ˙ y β = = R M = = ˙ ˙ 1 8 αβ β 2.15 α C ˙ β a ab ) a , α ab B ab , α −  ˙ R α ab e R R a β R β (Γ σ = 1 2 α P ) , + β + ) a a  ,  α e β , β σ ˙ ab + (¯ ) i γβ AB b ¯ e σ A , ω β , αβ + γδ − e , ˙ ? y γ α ? y M δ = ( e ab ∧ ab α ab α ? y + ( αβ ,R , , e ω y b y ) ˙ − ) defined in ( R b αβ α β a α M a a = 2 ˙ ) e α Ω := y δ ab e ω a ¯ (3) rotations are generated by ab R αβ y αβ γβ αβ σ ? αβ γδ ∧ ˙ ) ab , α σ ( ω αβ ω ˙ ) ω  + α so η i α R ) α a ab 1 2 γ 1 2 α α ) y ab e a 1 2 ) σ αβ ab a ω 2 = = = (¯  σ  (  a σ σ  ( de λ i † 4 i 1 ( σ β ( − = ( ) i 1 4 1 ˙ ( Ω + Ω h 2 αβ α − ˙ 1 2 + λ β 2 − ) d ) α 2 := λ − 1 8 α b αβ Y ) cd ab a = = := σ − a = := := = σ where (¯ dω R T ˙ ( ˙ σ α α ˙ Ω := α = αβ α R α (( αβ α = = = 3)-valued connection βα e ) ω  , abcd ab a a R R β  ab σ αβ (2 and the spatial R 2 1 M ( A R R 0 so 0 = 0 α In these conventions, it follows that and field strength The and raising andA lowering spinor indices according to the conventions using Dirac matrices obeying (Γ symbols obeying the oscillators λM JHEP10(2017)130 ,
s 1 is 3). µσ AB 4 > , η = 2 . 2 (0) B k (A.18) (A.19) (A.20) x g 2 p obeying AdS νρ principal SO(1 λ dX + (type D); ) defining A (0) polynomial π ∈ (0) } g 2 ... ω 2 s , S , dX − ) has multiple in the overlap + θ , 1 and [0 2 + ζ 2 . { 2 = ?L p νσ ) < ∈ 3 , 2 1 cos (0) S , + 2 ) (0) e − φ r 2 1 2 g x ), are related to the ds ) , p λx 2 1 ( = N is the self-dual part of µρ . If Ω( = generalized type D , 5.9 / ) θdφ L , as 3 ) ] is based on their alge- s µ S (0) s 2 < λ 2 2 s 2 1 as g x ] and λ where the gauge function 82 2 α = 0 (0) (type II); – = ( } − − ...α 2 , π 2 } λ 80 S N α [0 1 + sin = = . . . u of eq. ( s, s 1 , − T 2 ? dL { 1 α 1 ∈ ( µ α , µ N 1 1 ( t , = AB dθ x , µ 2 C θ x − ( η u { 2 2 L B = r ), x cos = 2 µ s X 2 spinors which one refers to as φ , X µν,ρσ 2 + = ∞ λ ) r x A , in integers obeying s s 2 2 (0) − ...α are the Riemann and torsion two-forms. (2 + s X [0 r sin 2 (0) 2 α R 1 2 2 θ α a λ ∈ bc 1 C − dr α = T λ vacuum solution Ω r c sin ) of 2 C µ – 59 – e s 1 + ) in which the metric reads p r 4 b ), , so 2 ) has,i.e., how many non-collinear principal spinors e ≡ s π + = = ζ 2 ) 2 ≤ 0 α s 2 2 , AdS . 0 ζ := k , we refer to the type (2 3 Weyl tensor. Clearly, this classification can be given in s [0 dt , ( s α 2 = 1 fixed. It follows that the single cover of ) s 1 ,X t, r, θ, φ a s 2 α 2 } C ∈ R T 2 µ r k 2 t x X ∈ 2 λ µ . . . u λ µ . The 1 and , related by the inversion x 0, and the transition function X leaves the metric invariant, maps the future and past time-like ab α t , X (type N) plus the trivial case of a vanishing Weyl tensor (type 2 , . . . , p η . This manifold can be covered by two sets of stereographic 2 ζ (1 + 1 , where λ µ ) < 4 b ν 1 } s p an arbitrary non-vanishing two-component spinor. Factorizing the e sin − 1 α 2 4 N,S 2 { X cd,ab a µ α { 2 u λx α ) φ , X e ( 1 + = r R ζ S AdS = 3). The stereographic coordinates / d 2 x = 0, with Riemann tensor , µ p e + ( cos i . . . ζ c x ) = 2 := 2 ds θ e 1 , of the five-dimensional embedding space with metric ζ (0) − ) 1 2 α is embedded as the hyperboloid i , λ 1 + λ µ ( Ω ζ A SO(1 . In the spin-2 case, this singles out the familiar six different possibilities: 2 µν Ω( sin ) 4 ) x / → − g ? s = p r X := +1 N (2 i µ 3) (type I in Petrov’s original terminology); µ x α p , = = (0) x ( AdS x viz. } ab C 2 1 0 1 > , λ R , (type III); and X X + Ω i 1 SO(2 p , } Petrov’s invariant classification of spin-2 Weyl tensors [ ) := 1 1 (0) ζ ∈ , , Ω 1 3 and { { O). The Type Dmassive case objects; is for related arbitrary spin- to gravitational field configurations surrounding isolated polynomial in terms of itsspinors roots defines a setroots, of the 2 corresponding principal spinorsdistinguish are how many collinear. different roots The Ω( classificationenter then the amounts factorization to of theterms spin- of the partitions braic properties at anymaking spacetime use of point. spinor language,Ω( Generalized it to amounts the to higher studythe the Weyl spin tensor roots context and of and the by degree-2 region The map cones into themselves and exchangeswhile the two leaving space-like the regions 0 boundary formally covered by taking providing a one-to-one mapthe if single covercoordinates, of are related locally to the embedding coordinates by in which The global spherical coordinates ( The metric d and vanishing torsion,L can be expressedcoordinates as Ω where JHEP10(2017)130 0 in (B.7) (B.8) (B.9) (B.2) (B.3) (B.4) (B.5) (B.6) . By (B.10) (B.14) (B.15) (B.11) (B.12) (B.13) † )) → z ( 2 ,  , δ } † b P ) ? ∓ α v {z times ) = ( ··· z n (¯ := ( , 2 . b , , i 2 ˙ P? | δ  α ) (B.1) ) ¯ z z ¯ v y z − ¯ κ = ( ( 2 2 = δ and δ ?n b b F ? κ F? , b − , α † . P , , , ,, , M M v ? ? ) − ˙ ) − ) ˙ 1 )) α ) 1 α α z ¯ z z y ¯ z y y y ¯ z ( + (¯ (¯ ˙ ( κ κ ˙ α det ? η det 2 ? ζ α 2 2 N 2  ) ,  δ + +  v πδ v πδ πδ η ζ ?n ) = ) = ) := ) := ¯ + = ¯ = ¯ b b b P , v F F , = 2 := := = 2 † = 2 π ! † ( ( , it makes sense to define the following ) ( 1 ) ) = ( α α My z Mz ¯ n y z z ¯ z )): y z i R ( y ( z ∓ y ¯ ¯ ∓ ( N ? N ? 2 C (¯ 2 α η α ? ζ R 2 =0 ∞   1) 1) δ X n v v − − exp := ( := ( = – 60 – , κ 0 ,, π π = = ,N ,N ) GL(2;  ¯  y y b , δ , δ = ( → = ( P y z  ¯  y ¯ η  ζ ¯ κ ) ) ( )-invariance is restored in the limit † ? ∈ † N ? N ? 2 2 ) ) 2 = lim y C z β y z α 1) 1) ( ( , b b κ κ y πδ F ? κ F? exp δ δ − − M ) ) N ? , , ? ? 1 1 ,, η ζ ¯ y z y y − = 1. The inner Kleinian elements generate the involutive 1) − = 2 := ( := ( ( ( ¯ ¯ := ( := ( η ζ κ κ ? − δ δ y ¯ ¯ z y , ] ( z ? y ? = 1 = 1 κ ¯ ¯ κ κ ) + κ κ c ? ? + ¯ + ζ ] ] ) := ) := ¯ ¯ η ζ , ) as ) := ) := + + b b − F F z y C log y = ¯ = ( ( ¯ ( ( ζ , ζ , η ¯ 2 y y 2 N ? b † † ∈ P − − δ δ ¯ π π ) ) ( ζ 1) η c y z = [ ? into a complexified Heisenberg algebras [ [ − ? N N α ] z + ¯ η := ( := ( = exp , . The broken SL(2 z ¯ ¯ z viz. y and − b P ? ¯ ¯ N ? η c N N α 1) y − ( such that [¯ automorphisms using We also define and representing ( idem Weyl order, using the notation ( one can define idempotent inner Kleinian operators Their hermitian conjugatessplitting are defined by Working with the chiralcomplex integration analytic delta domain functions ( B Properties of inner Klein operators JHEP10(2017)130 z )– π y viz. π (C.1) (C.2) 3.16 (B.16) (B.17) (B.18) = , π )) ), that de- z (¯ 2 δ 3.25 ) y )–( (¯ 2 variables in ( δ 2 Z ) 3.21 π and ((2 ). Note that the deformed Y . , . Weyl 3.17
) ) . z ¯ z z O ¯ z )–( z, ? κ ? κ ? κ − ) = ¯ ¯ ; Ψ y Ψ ¯ b κ 3.15 b ¯ ¯ b y ( κ . The inner automorphisms − y, − Z = 1 − ¯ b κ (1 ( b Normal ˙ . It is then possible to move back the so- f β and ˙ ]) for a constant deformation parameter. We αβ α O Z 3 i i Y – 61 – , 2 2 )) = z ¯ − − z , z, ? κ = = ; )) y ? ? ), and of the separation of ¯ y z ] ] κ ( ˙ 0 β 0 β 2 y, within the Ansatz ( b ¯ S b S δ ( = 3.15 , ) , b f ˙ 0 Z α y b κ 0 α ( ( b ¯ S b S π 2 [ [ δ on 2 0 ) α π V ((2 Weyl ] for a detailed proof). 1 O , however, act locally on symbols defined both in Weyl and normal order, ¯ z ] (and, in a different gauge, in [ ) = ¯ π 1 ¯ y b κ ( π = ¯ ), the deformed oscillators obey π Normal O 3.17 where the right-handsolved sides in have [ acan distributional use deformation the same term. solutionas method we This by shall replacing describe problem the next. was constant deformation parameter by Ψ, As a consequence of the( choice ( appendix G in [ C.1 Problem setting In this appendix, wetermine recall the connection the mainoscillator steps problem of with the a solutionone, distributional with of deformation a eqs. term regular ( canordering Gaussian be source, used reduced upon here to changing toobtained an ordering solution normal prescription, ordinary to from ordering Weyl the ordering on Weyl and get the same result that we shall review here (see C Deformed oscillators with delta function Klein operator providing an example of theorder fact and that completely one entangled and in the another same order. operator can be factorized in one This action is generated by conjugation by the elements Their Weyl-ordered symbols can be read off from order that depends non-triviallyand on ¯ both that act locally on symbols defined in Weyl order, but not on symbols defined in normal JHEP10(2017)130 - Y (C.9) (C.5) (C.6) (C.7) (C.8) (C.3) (C.4) ) , + , z z -product ) ◦ z 0− ρ ? κ , − ? κ ) Ψ , − i b z ) 2 Ψ =0 . + + b 1], while the 0 − 0  − z ρ , ρ ρ =0 w

1 − + − α 1 ) = z ρ ρ − = 1 +1 − i +

2 w ? ˜ ˜ t t (1 z ) are star-functions of ] ) i i z ? + t 1) β − e | 2 + w  z ρ − 0− 0 1 z z β − t ρ − , z  ρ +1 − ( w (Ψ w 0 0 + − z ( 1] to [ 1 − 1 +  ρ = tt tt z , z +1 − z +1 − w i i 2 t t f 0 + 0 i +1 1 w t t e 0 i 1 ρ ? − t ,[ i , ] e − 1) + e + [  − + z − + 1 z − α z 0 0− z t and + w ˜ ˜ t t ? e − ( b × S w ˜ + i ρ ( tρ 1) ? z , + α 1 ) z − ? z 0 1] − i ) u +1 t t + 0− − t , t = 1) | ( ) z + 1 − | 2 z − β b 0 1 + 1 ( S := 0 0 − w t [ + u − α + 1 | 1  ρ − z (Ψ ρ i tt tt (Ψ +1 0 u ? e +1 − ]) and adapted to a distributional defor- t t i − +˜ ?  t t w  ) + 1) + (Ψ z i t f := , 0 f + β α 67 ) e | w + t 0 α u , u ? e 2 ( t f − ? 1) 1 + z  | ) − α ) ˜ (Ψ ρ  − α , t sends [ ) ˜ t u | ), the latter turns into the integral equations  (  – 62 – z ? z (Ψ + ? 0 dt ( 2 f + 1) z u w Y, z , ) + = tt (Ψ ) ( 1 t t − i 0 +1 | C.3 ); ( + ρ ˜ +1 t −  t t t z | 0− := ρ αβ ? f − α i f dt + 1 ( 0 + 1) 7→ to split ( b  (Ψ S t e ) − 2 t  ˜  t ] (see also [ ? e t (Ψ z 2 )  + α |  α z 0 u − 1 + + 1 + f 2( πδ u ) can be obtained explicitly by employing the u ) dt + 1 66 1 ρ − 0 1 ˙ 0 α 2 (Ψ ) into ( + t t, t t − + Z ? z 0 − | ¯ + 1) ? f z b S − Z ρ 1 = 2 ) 1 ) 0 α , 2 w ) ) as ( t − (Ψ C.4 t 0 z if | ∂ α dt tt | 1) + 1) Z 0 b := ( ∂ρ  κ S − = 4 t − Y, z ¯ b t S (Ψ α (  ( (Ψ + 1) ( + , 0 α + 1) ( ∂ − ˜ ? f  = ρ t + 1 − 1 b 0 ˜ S t ( ∂ρ f 0 ) − i f b +1 α S iu i t b t  S Z | 0 2( 4 4 − α  dt u − − u (Ψ , ? e ]. The method can be articulated in the following two steps: 1 1 0  1 = = ≡ = 4 − ) tt f are classical sources, t ) = Z | ], later refined in [ z 0 α | = dt b := ρ S 64 + 1 (Ψ Y 1 ˜ t 1 t (  − 0 α f Z b S 4 Thus, upon inserting ( where we recall that where the induced mapdependent ( pieces behave as spectators. In particular, star-product, as can be seen from where Ψ. The virtue of these generalized Laplace transforms is that they are closed under and represent ( We introduce a spin-frame i) The deformed oscillators ( method of [ mation term in [ C.2 Integral equation JHEP10(2017)130 z κ − = 1); (C.18) (C.14) (C.15) (C.16) (C.17) (C.10) (C.11) (C.12) (C.13) g ) = ◦ z are given , κ 1) ), changing (  k , 2 − τ λ ( t ) ◦ C.9 z g ( , log 2 ) = 1, 0 Ψ 2 z ? δ tt b and Ψ g − ?κ − , z u πb ) = κ ( ;2; t 2 | , δ 1 2 , z ) − 0  , (Ψ t Ψ (+1) 2 1 | z j ?κ b ◦ F ) w g . 1 = 1 1 z (Ψ − , 2 +1 − ? on the space of functions on the z Ψ t t 1)+ − 2 z i b ( w 1) Ψ − 4 1 27 f ? h -product algebra; and b t σκ +1 − . − ? e − ( ◦ u u ) δ − ) i  t = u t | 1) e ) = ( | f z z δ  ◦ − ( w ) = ) = (Ψ (Ψ z t t ε 1 1 t ˘  | | ˘  1]) into the left-hand side of ( ( w h ?f , 2 1 iσ 2 ) = δ 0 t 1 1 z (Ψ (Ψ − u  | κ e dt -product equation − dt − − + 1 [ – 63 – ε ◦ 1 1 = ) = 1 1 u u t 0 − ∈ )(Ψ | , q i  1 Z 0 1 ) + → t f − lim ε ). Next, using the delta function sequence (Ψ f  ( Z dt t, t δ u k h 1 + contains gauge artifacts follows by using holomorphic ◦ | 1 1 p  )( − g − 0 ? ) = Z f )(Ψ tt ) ( (Ψ) + -product composition to accommodate functions of oscillators, it is of k u − | ◦ Y, z f λ u ( ) := ( ◦  (Ψ u =0 ). | ∞ ) act as projectors in the h k − X t f , f ( f 2 ) du δ ) is compatible with ◦ )(Ψ k )+ 1 ( C.44 t 2 -dependence comes through one and the same function, Ψ in this case. 1) 1 | δ − ) is equivalent to the h  Y du R + 1) ( k C.12 ◦ ) := ( ◦ (Ψ ! u 1) u g q 1 k C.9 ( . The fact that | − h b ( κ 1 = 1 ( is a gauge artifact (and we use the notation ) and ( (Ψ − ) =  t = g Z h | f 4 z C.41 ) := (Ψ t j ( ? κ k y where p by ( with the following solution space, as we shall show in the next subsection: Thus, eq. ( we find the unique solution unit interval, one arrives at where order of integration, and defining which is a commutative and associative product κ Inserting 1 = For this generalization of the ii) 27 which induce course crucial that the The presentation ( and gauge parameters of the form JHEP10(2017)130 ) C.22 (C.30) (C.24) (C.25) (C.26) (C.27) (C.28) (C.29) (C.19) (C.20) (C.21) (C.22) ) and ( . 1 ) C.21  k s . = ) 0 s ··· | 1 ) , s C (Ψ ) t , f − ( ∈ k ) , k u π ξ ( ( k , 1) Ψ 2 δ p , π, π , 1]) b − ds s ) , Ψ k 2 k s 1 1 b 1 ξ − , ( − (Ψ) ds ) − = ( ) ) [ π Z − + 1)] (C.23) 1  k u u 1 ( ( ( λ − , (Ψ) u , ∈ ) ) 1) , ( Z l ) πf t π − δ ?k ( =0 + 1 (+) ( π ∞ 0 0 − ( k k X I I  k − ···  I u )Ψ k , π µ ](Ψ) = 1 ) = t (
1)! ) ( s f δ 1) = 2 ) + ) s [ 1 ) = ) = ds =0 t u ∞ − ( ) − π k | k X u u ) − ( 1 ( k π 1 | | k k ( ) l I − ( ) = ( u π I (Ψ – 64 – log Z δ ( ( u )  k | ◦ δ k π )(Ψ )(Ψ ) := µ [ -product subalgebras. Equations ( ( ) ) ) ) ξ ! ◦ 1) π π |  π − 1 2 ,L 0) k ( k )(Ψ =0 ∞ − − ) − I k m X +(+) +( + ( π (Ψ (1 (1 > ( k := ) f f , f f 1 2 1 2 -product algebra decomposes into even and odd functions p π ) ) k ] ◦ ◦ ( 0 ◦ ◦ )] )] ) = ) =  π f )  t t ( u u ( 0 [ | | ) := − − I k e g s (+) ( m f ( L ( ◦ − − (Ψ (Ψ ) ) ) ) π f f ), i.e. π π k ( = ( ( π ( ( ( p   f f viz. 0 = [sign( f ◦ m C.14 ) := [sign( ) ππ 1], δ π u ) implies that upon defining the symbols ( ( k 0) , ( p ) 1 = 1) π > ( − k ) ) separates into the following two independent equations: 0 I C.27 > π ( k, l k g -product projectors ( C.19 ◦ ◦ ) π ( f The property ( and of the obeying which obey ( in terms of ( acts as the identity in the evencan and be odd cast into algebraic equations by expanding ( where Therefore ( In order to solve eq. ( one begins by observing that the on the interval [ C.3 Solution using symbol calculus JHEP10(2017)130 ): (C.36) (C.37) (C.38) (C.39) (C.40) (C.31) (C.32) (C.33) (C.34) C.25 . . 1)   2 ( 1 s ? . Thus, substi- )) , ) ξ . log | ) π ( + ξ , ( Ψ e 2 (Ψ m ) ) b ) = 0 − ? − − ) ( ( ; (C.35) 0 ), one is left with the ( ) e π ε I ) e ; 2; g π , π ( − ◦ +( k 1 2 ( k 1 e ) λ  m ) = C.29 ], the algebraic equations involve − 1 − ξ ? λ ξ , ( ) = ( | = ) F ε ξ ξ 67 , | . ) 1 π Ψ 2 ( (Ψ π b = Ψ ( ) 2 ? − k 66 (Ψ b , )  k − ) λ − ( , = 0 ) and ( − 64 − Ψ ( 4 − ( e m  b ]  1 ] + ) ) − π e C.27 π m r ( ) = 1 ) = 1 ( , λ = ξ ξ − | | m ) = [ 1 π , m k m ξ , ( k k [ − (Ψ (Ψ ) ξ µ L ) k n Ψ 2 )  1)!
ξ , − b π 2 ) is parameterized by an undetermined function = ( k 1 s – 65 – + 2 ?L − +( +(+) (+) ) µ Ψ − 2 ) ) q b π e e k m m ( 1 π π ( ( k =1 log ∞ C.32  k ) contains an undetermined set of coefficients, say + ? ? − k X λ +( k r µ ), and using ( ) ) 1  λ ? ξ ξ ?k | | (+) 0 ) ? r C.33 I ξ ) ] + ?  (Ψ (Ψ ( C.22 ) = referring to the spin-frame. Thus ) ⇒ (Ψ) π ) and ( ) ( k ξ π ◦ k | 1) − λ (+) (+) (   +(  e ε ( − − (Ψ f (+) b 2 ? C.31 is mapped to the algebraic product m e e ε [ m m )) ) − = n ) and ( (+) ) = ξ π = e | ( + q ξ  | ?L f m   (Ψ ) (+) 1 2 C.21 k (Ψ ◦ π ( )  m (+) = 1, and π − k (+) e g ( − 2 λ =1 e ∞ ? m ) has the symbol 28 m k X s )) | ) into ( ξ ) = ( ( ξ ) ) = (Ψ | s  | ( C.25 (+) e (Ψ ε q (Ψ . One can show that the undetermined quantities are gauge artifacts. A natural -product (+) as follows: ) ◦ Note that, differently from the Lorentz-covariant solutions in [ π (+)  ) q π 28 e k +( m ( σ e the product of two different functions rather than the square of a single one. where corresponding to the confluent hypergeometric function It follows that where ( λ gauge choice is to work with the symmetric solutions henceforth we shall drop the g Likewise, the solution space to ( and the following condition on the coefficients of the projectors in the expansion ( The solution space to eqs. ( tuting ( algebraic equations the JHEP10(2017)130 ]. Phys. , (C.41) (C.42) (C.43) (C.44) (C.45) ] J. Phys. , SPIRE IN ][ need not vanish, k ) in the symmetric p , C.4 ), there still exist non- , arXiv:1107.1217 , # ] [ 2 ) / . s C.2 1 (+) ( } ? ) q 1 [ )–(  − ) into ( , k ( 0 arXiv:0906.3898 ]. k ]. 0 I L Ψ C.1 b . C.25 (2011) 084 1 + ) + ∈ { s SPIRE SPIRE ( − k 12 IN = 1 + = 1 . IN ]. 1 [ -product projectors i k ) [ (+) 0 ◦  π I ) and ( ( 0 (2013) 389] [ (+) Z − of q JHEP , θ SPIRE 1 ] , k? C.40 " ) 1) = IN λ π , giving rise to a non-trivial flat connection on k ( )-invariant, non-trivial vacua. In this paper, for ][ − – 66 – 1) + , µ k = 0, that is B 722 (1988) 491 , we have p (1990) 378 θ 1) m s , deferring the study of the non-trivial vacua to a k k [ 2 − ( θ p k k − − δ 2 s = 1 − L ( 4 ) Static BPS black hole in 4d higher-spin gauge theory k δ = =  θ h p, B 209 ( B 243 1 + ( 2 k ε =0 − ), which permits any use, distribution and reproduction in k L − θ Families of exact solutions to Vasiliev’s 4D equations with Biaxially symmetric solutions to 4D higher-spin gravity Ψ=0 | | = ) s Erratum ibid. ) ] = ] = k? | = 0 for all π [ λ ( k m (0 k ). arXiv:1208.4077 [ (+) λ θ k  [ for Ψ = 0 and q Phys. Lett. [ Phys. Lett. f L , k α , 3.30 z CC-BY 4.0 L Consistent equation for interacting gauge fields of all spinsEquations in of motion of interacting massless fields of all spins as a free are Ψ-dependent only for even = This article is distributed under the terms of the Creative Commons ) ] for similar, SO( (2009) 305 0 α π ( k b S 67 λ ]. ), yields ( (2013) 214004 ) and B 682 SPIRE C.35 C.41 IN Lett. (3+1)-dimensions differential algebra spherical, cylindrical and biaxial[ symmetry A 46 M.A. Vasiliev, M.A. Vasiliev, C. Iazeolla and P. Sundell, V.E. Didenko and M.A. Vasiliev, C. Iazeolla and P. Sundell, ]; see also [ 1 [ [4] [5] [2] [3] [1] Attribution License ( any medium, provided the original author(s) and source areReferences credited. Z simplicity, we shall take future work. With thisgauge choice, ( insertion of ( Open Access. C.4 Non-trivial vacuum connectionsFor Ψ on = 0, eventrivial though vacuum solutions, the as deformation the isbut coefficients absent rather from they ( reduce to it follows that implies that From ( Requiring that As for the coefficients of the projector JHEP10(2017)130 ]. ]. , (2016) ]. (2006) J. SPIRE higher , , IN 09 [ SPIRE D ]. IN 4 ]. SPIRE ][ B 752 IN JHEP ][ , SPIRE SPIRE hep-th/9910096 IN (2014) 044 ]. , IN arXiv:1512.07932 ][ ]. , ][ 11 ]. Nucl. Phys. SPIRE , ]. ]. arXiv:1401.2975 IN SPIRE , [ JHEP IN SPIRE arXiv:1205.3339 , ][ IN ]. [ ]. ]. SPIRE Nonlinear higher spin theories in SPIRE ][ arXiv:1512.02209 IN IN [ [ ][ An action for matter coupled higher hep-th/9611024 SPIRE SPIRE ABJ triality: from higher spin fields to [ SPIRE (1978) 421 arXiv:1102.2219 IN IN [ IN [ 2 [ (2012) 043 ][ ]. 10 (2016) 003 A minimal BV action for Vasiliev’s (1991) 1387 hep-th/0103247 (1996) 763 SPIRE [ 05 8 – 67 – (1989) 59 IN arXiv:1207.4485 (1992) 225 [ ]. 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