JHEP11(2018)167 Springer , November 9, 2018 November 27, 2018 September 14, 2018 : : : Accepted Received Published a Published for SISSA by https://doi.org/10.1007/JHEP11(2018)167 [email protected] , and Euihun Joung b . 3 Xavier Bekaert 1808.07728 a The Authors. c Conformal Field Theory, Higher Spin Gravity, Higher Spin Symmetry
The linearized spectrum and the algebra of global symmetries of conformal , [email protected] [email protected] Department of Physics andKyung Research Institute Hee of University, Basic Seoul 02447, Science, Institut Korea Denis Poisson, Universit´ede Tours,Parc Universit´ed’Orl´eans,CNRS, de Grandmont, 37200 Tours, France E-mail: b a Open Access Article funded by SCOAP ArXiv ePrint: ization adapted to theirThis evaluation result holds and in observe thetheories that case and of their type-A confirms characters and previous are type-Bdegrees observations (and actually of on their equal. freedom higher-depth a in generalizations) remarkable conformal rearrangement higher-spin of gravity dynamical after regularization. Keywords: Abstract: higher-spin gravity decompose intobra. infinitely many Their representations characters of involve the divergent conformal sums alge- over spins. We propose a suitable regular- Thomas Basile, Conformal higher-spin gravity: linearizedsymmetry spectrum algebra = JHEP11(2018)167 ] and 1 12 11 41 5 37 37 15 9 5 43 theories ` – 1 – 16 19 21 theory ` theory theory and type-B ` ` 34 ` 7 26 1 23 C.1 Branching rule forC.2 generalized Verma Branching modules rule for irreducible lowest-weight modules 3.3 Character of type-B 2.5 Character of type-A off-shell conformal higher-spin gravity 3.1 Field theory3.2 of type-A Character of type-A 2.1 Field theory2.2 of type-A conformal Relevant higher-spin modules 2.3 gravity Character of2.4 the Fradkin-Tseytlin module Character of type-A on-shell conformal higher-spin gravity usual diffeomorphisms is a challengingmany question interesting and results attempts not at only answering toold it the example have physics brought is of provided gravitation by butconformal the also geometry. to fruitful mathematics. exchanges Higher-spin between An Weyl’sgeometry gravity theory remains is of somewhat gravity a elusive. [ much However, younger its deep example connections and with its important underlying topics 1 Introduction Whether Einstein’s gravity can be extended to a theory with larger symmetries than the D A primer on nonlinear CHS gravity B Two-dimensional case C Branching rule for the Fradkin-Tseytlin module 4 Discussion A Zoo of modules 3 Extensions to type-A 2 Type-A conformal higher-spin gravity Contents 1 Introduction JHEP11(2018)167 – 17 ] and with 7 – 5 ]. This holo- are dual to the 31 , +1 30 d -forms, respectively. 2 ]. In three dimensions 2 − d ], where the higher-spin 16 , jet bundles, tractors, etc) 10 15 1 ], which can be computed perturba- 29 – 27 – 2 – ]). of free scalars, spinors and ] for early works on the CHS superalgebra) where 26 d – ]. Aside from this seminal paper, other examples are the relation 14 20 2 , ] where the effective actions have been calculated in the world-line 13 35 , while the higher-spin holographic duality ensures that there is 34 ]). This scenario actually extends beyond the case of the con- 2 29 ], etc. 9 , 8 ], the connection of the latter equations with Fedosov quantization [ ] (see also [ 4 , 3 The type-A, -B and -C higher-spin gravities on AdS ], as well as [ 12 , 3 33 11 , 32 CHS gravity shows several interesting properties. In particular, it shares the main In fact, one of the simplest ways to understand CHS gravity in even dimensions is What we shall study in this paper is the theory which combines the conformal and See e.g. [ In the sequel, the somewhat ambiguous terms “CHS theory” or “CHS gravity” will stand either for the The close relationships between higher-spin gravity and deformation quantization was observed in 2 3 1 ]. The unfolded formulation on which Vasiliev’s equations are based can be also applied nonlinear equations [ some formality theorems [ formulation. specific example of type-Amake CHS it gravity clear. or, on the contrary, for generic CHS theories. The context should ory so it is non-unitary. On the othervarious hand, instances it over the is lastspin an algebras three in interacting decades. terms theory Itbetween of was with the star first products quantized a observed [ hyperboloid by massless and Vasiliev in the his deformed realisation oscillator of algebra higher- which is instrumental in Vasiliev the zoo of CHSMHS theories following theories. the by-now standard(singlet sector nomenclature of) (type A, vector B, model etc) CFT for drawback and virtues of Weyl’s gravity. On the one hand, it is a higher-derivative the- of MHS theory (seeformal e.g. scalar [ field, thethere construction is of a CHS one-to-one theories(CFTs) and via correspondence CHS the between theories, effective freea action vector one-to-one ensures correspondence model that between conformal them and field MHS theories theories. Accordingly, we will denote can be again relatedthe to MHS (the gravity logarithmically in divergentfield part one equations of) higher can dimension the also with on-shell beFefferman-Graham) bulk standard of obtained action boundary as the of the conditions. solutions obstructiongraphic to The in picture the the CHS makes bulk power it MHS series clear field expansion that equations (`ala the [ global symmetry of the CHS theory matches that to free CHS gauge fields (see e.g. [ viewing it as the logarithmicallyfield divergent in part of the the background effective oftively action in all of the higher-spin a weak field fields conformal expansion. scalar [ In the context of higher-spin holography, the CHS action cubic interactions of CHSformulation fields [ were analyzedthe by quartic Fradkin order and analysisconstruction nevertheless Linetsky becomes of in the difficult, a massless similarlyhowever, frame-like higher-spin to the the (MHS) full non-linear Fradkin-Vasiliev interactions action [ could19 be obtained in the Chern-Simons formulation [ might keep some surprises in store. higher-spin extensions of gravity, namelyof conformal CHS higher-spin (CHS) gauge gravity. fieldsgeneralization The was of study the started linearized by conformal Fradkin gravity and action Tseytlin was in obtained. [ Subsequently, the of modern mathematics (such as deformation quantization, JHEP11(2018)167 ]. As 40 ]. Unfortu- 38 dimensions and has d ]. 37 , ] for related discussions). ]. 36 45 44 – 42 ] for other strategies). 50 , (see also [ -anomaly would also vanish assuming a 49 4 c ] field is local in even ] for the original arguments). In fact, an s 41 10 – ]. There exists yet another approach to CHS 38 -anomaly vanishes or not, one needs the linear – 3 – 52 c ]. The , 51 ] (see also [ 46 , ]. The twistor CHS theory has the same linear action 48 38 53 , 47 ] in the context of conformal gravity). 56 ) as was suggested in [ d equation in an arbitrary gravitational background, which has been the ] (see also [ s 55 , 54 4 derivatives. Generically, the corresponding kinetic operator is higher-derivative − -anomaly coefficient vanishes [ d In this paper, we will focus on the spectrum and symmetry algebra of CHS gravity. Scattering amplitudes of CHS gravity also exhibit surprising features: four-point scat- The linear equation for the conformal spin- a The factorization in the spin-two case was observed already in [ + 4 s the full interacting level. Recently,studied scattering in amplitudes [ of twistor CHS theory have been More precisely, we will show that its total space of on-shell one-particle states in even tering amplitudes of external scalarthe fields, infinite Maxwell tower fields of and CHStheory Weyl fields using gravitons vanish the mediated [ twistor by formalism [ as the conventional CHS, but it is not clear that the two CHS theories are the same at factorized form of thenately, equation such in a the factorization presence doesa of not consequence, non-vanishing actually to curvature happen determine [ inconformal whether general, spin- the as was shownsubject in of [ several recent papers [ 2 and can be factorizedspin into (around the ones AdS forThis massless factorization and allows partially-massless usthe fields to of the compute same easily its Weyl anomaly and it was found that metric-like formulation. Moreover, itIn appears fact, non-standard to MHS possessimposing gravity similar could suitable features possibly boundary to conditions arise MHS aroundsame from an theory. way CHS that anti gravity de Einstein’s at Sitter gravity tree is (AdS) level related background, by in to the Weyl’s gravity [ fact, due to theremarkable presence cancellation of properties huge in symmetries, their suchtition scattering systems amplitudes, functions, have one-loop etc. been anomalies, From par- observedas a to a higher-spin exhibit theory of theory interactingIt perspective, massless has CHS and the partially-massless gravity nice fields can of feature be all of viewed depths being and perturbatively all spins. local and of admitting a relatively simple suggested by some preliminary tests.the study From a of quantum CHS gravitytheory gravity perspective, of as this quantum an motivates gravity. interestinginvolve toy infinite The model towers spectra of of of gauge arequire higher-spin fields. UV-finite some gravity The (albeit theories regularization corresponding non-unitary) (CHS which infinitesuch and number infinite opens MHS) of collection the contributions of possibility fields that may the be collective much behaviour softer of than their individual behaviour. In coupling, so it issymmetry a thus theory it of isgauge gravity. anomalies scale-invariant Moreover, (in at its particular, the globalsure Weyl symmetries classical anomalies that include and level. CHS their conformal gravityadvantage higher-spin is of Consequently, versions) CHS UV-finite the gravity would (see over absence en- its e.g. spin-two of counterpart [ is that it might be anomaly-free, as spin-two field in its spectrum whose interactions with the other fields include the minimal JHEP11(2018)167 ` ) ) ) Y is 2 d φ , d Y 1.1 (1.1) (1.2) ] and (2 so (which is 52 and for , , . . . , x 1 Y 51 x and spin 3.2 CHS theory ` ] and type-B = ( PM 31 [ ) to a couple of x ` , 1.1 ) x in section and , ` and same spin 1 ], then move on to the q ) relates the characters . − by discussing our main q PM 67 1.2 ] of type-A CHS gravity. ( 4 , ]. ∆ 63 29 66 ] for related discussion), then − , – ) cancel each other because of (P)M This identity was first derived d 46 χ 65 . The idea of the proof of ( 58 1.2 , 5 − with minimal energy ∆ 39 A ) +1 x , we start by reviewing the field the- d ) characters of a Killing tensor and its q, 2 ( , of the associated partially-massless field , d d ) angular momenta provides an alternative , we extend property ( (2 ] (see also [ d in AdS are isomorphic to the conformal Killing tensors of ( (P)M ) holds for the type-A 3 so 64 Y χ so +1 ϕ – 4 – 1.1 d ) + x with conformal weight in the generating functional of correlation functions of the dual d q, in AdS ( Y , Y ∆ ϕ J on M CHS ). In section χ Y ) over all fields in the spectrum of CHS gravity. After the designate the φ 1.1 1.2 . We conclude the paper in section ) = x ) in the next section). The identity ( (P)M On-shell CHS = Higher-spin Algebra χ ]. The generalized version of the identity to partially-massless as q, 3.3 ( 65 2.11 KT and χ ] for the analysis in full generality, as well as [ CHS and move on to show that ( 57 carries the same (reducible) representation of the conformal algebra in section , χ as well as the Killing tensor of the latter. , ` d 21 ) might be that the dynamical degrees of freedom of CHS gravity reorganize 3.1 KT +1 1.1 d χ . See [ d The paper is organized as follows. In section Let us recall that any partially-massless field 5 that turning on chemical potentials for the associated with a conformalused as field a source forCFT). the Moreover, dual the operator Killing tensorson of M theories. We start by describingin the section field theoretical realization ofthe the type-B type-A result and commenting onfunction and its the implication free energy for on AdS the background computation of this of theory. the In particular, thermal we partition point out oretical definition of type-Aderivation CHS of theory the as propertyclasses spelled of ( out higher-depth CHS in theories, [ ciated whose to spectrum the are partially-massless made fields up making the of spectrum the of CHS the fields type-A asso- parity properties of the correspondingresult characters. ( A possible physicalinto interpretation its of asymptotic the symmetries. Ininterpretation this resonates sense, with similar CHS observations gravity on isthe the somehow one-loop scattering “topological”. partition amplitudes This [ functions on various backgrounds [ for the massless totallyfurther symmetric explored case in in [ [ well as mixed-symmetry casesis is to perform derived in thesum, sum appendix the of terms corresponding ( to the last two terms in ( where associated CHS and (partially-)masslessare field, related respectively. to Here, thethe temperature usual and way (see theof ( chemical a potentials given CHS for gaugein the field AdS angular on flat momenta spacetime in M spin algebra, which is isomorphicIn to such the higher-spin derivation, algebra it of is the crucial associated to MHS use gravity. the identity as its globalalgebra), symmetry i.e. algebra (spanning the adjoint representation of theStrictly higher-spin speaking, we willof sum CHS gravity the and characters compare of them the to the CHS character fields of making the corresponding up conformal the higher- spectrum dimension JHEP11(2018)167 . s 1 − d (2.2) (2.3) (2.4) (2.1) S × a rank- 1 . In ap- S d 2 contains spacetime − s 6 A σ field. Due to , we provide a /CFT ) satisfying the compensates the 2 on-shell Fradkin- +1 D d − 4 s 2 − d shadow , σ ). + 1 s contains the branching − s 2.2 hereafter. An equivalence C ξ where equalities that only , ≈ s , h 2 4 (FT) field is a symmetric rank- − 2 , − . s d is the projector to transverse and 0 + -dimensional Minkowski s η σ = 0 d s TT ≈ 2 P + s the Minkowski metric and − h 1 s s TT η − 4 Bach equation P s 2 ) reads − s η σ d s ∂ ξ + 2.1 + s . It is also called as the x h + 1 s d – 5 – should stand for its conformal compactification d − s d s − field in even h tensor and d s TT s M P s ∂ ξ ∼ . The differential operator is the set of parameters ( Z s = 2 s h s h ] = s off-shell Fradkin-Tseytlin h [ ), together with a more detailed description of the various obeying this equation will be referred to as FT 1) symmetric tensor, 1.2 S s h − = 2 case as a toy model, while appendix s , the action has the gauge symmetry d s TT so that the action is local and conformally invariant. P ] and provide a heuristic argument supporting the “topological” nature , referred to as 28 s TT s a rank-( h P ) of fields field. 1 − 2.2 s ξ conformal Killing tensor 2) symmetric tensor. Here, we used the schematic notation where all the indices is a totally symmetric rank- we detail the − s B h s This equation is referredhold to on-shell will as be denoted the by spin- class the weak ( equality symbol Tseytlin i.e. the gauge parameters leaving the FTThe field equation inert of under motion ( of the action ( ( are implicit. The conformal Killing equation field of conformal weightthe ∆ projector with The field Whenever global issues would be relevant, M • • • , is described by the local action 6 d where traceless symmetric tensors ofnon-locality rank of 2.1 Field theoryThe of free type-A theory conformal of higher-spin theM gravity conformal spin- short review of nonlinear CHSproach theory. of In Segal particular, [ we reexamineof the CHS formal gravity. operator ap- 2 Type-A conformal higher-spin gravity pare to the previouslythe used derivation regularization of in identity themodules ( literature. of importance Finally, appendix and theirpendix field theoretical interpretations inrule AdS of the on-shell (totally-symmetric) CHS field module. In appendix regularization of the sum over the infinite tower of higher-spin fields, which we also com- JHEP11(2018)167 via (2.7) (2.8) (2.9) (2.5) (2.6) s ): (2.10) of the h φ . Note log , log ] is related s W κ W h . Moreover it 4 derivatives. [ s = 2 and is in- = h FT − S s,s d = 2 it corresponds C CHS s S gauge fields. An in- : of the FT field (here , , s κ s,s s , C h , ) 4 s ] 3 s ) at the quadratic order: . 2 N − J d h ∈ s ( 2.1 } =0 ∞ s O s X should be equal to its AdS coun- h s s { s,s ; + φ Weyl tensor ] + ] in a higher-spin background. Start- [ log coupled to higher-spin sources φ s S x C h W − [ φ ). In particular, for d 67 denotes the complex conjugate of , d ¯ φ FT ¯ φ e φ d 2.2 1) S 34 − D M , ), which can be perturbatively calculated — x Z ,d 3 s can be regarded as an action of interacting FT ∼ Λ d =0 ∞ s X h (1 Z d s 29 – 6 – ( , d so log 1) = O (where M 27 ] = ] W − Z N N φ ] + regular or polynomially divergent terms ∈ ∈ s s s s,s s } } h ] = s C ] = ( [ ¯ s φ ∂ h s N . Schematically, we have { h [ ∈ h FT is the (generalized) { [ s = Λ [ +1 } d s s W s FT J log h ) corresponds to the conformal scalar with is a primary field with conformal weight ∆ RS − h S s e { W ]) to reproduce the free action ( ; ]. Therefore, ∂ 2.9 s,s φ [ 67 C s,s T 67 S , P ) can be also rewritten, after integrating by part, as ] = log is a local and nonlinear functional of the shadow fields s 48 , = 2.1 h Λ 29 K W s,s = C ) below). It is a traceless tensor with the symmetry of a rectangular two- = 4 the scalar field becomes auxiliary and drops out from the spectrum of CHS. s = 0 term in ( denotes the traceless projector onto the two-row Young diagram displayed d [Φ 2.6 s s,s T to the linearized Weyl tensor. row Young diagram, The Weyl tensor variant under the gauge transformations ( where P in ( The action ( According to the AdS/CFT correspondence, So far we have considered the free theory of conformal spin- Fronsdal • S terpart, that is, thewhere logarithmically the divergent part location of oftheory the is the on-shell linked boundary AdS to action is theto in type-A pushed the the MHS Fronsdal to theory. limit action infinity. For in instance, AdS In the free this action way, the type-A CHS see for instance [ fields, up to the introductionthat of the a dimensionless couplingHence, constant for can be shown (see e.g. [ and contains also the interaction terms where Λ is theeffective action ultraviolet (UV) cut-off. The logarithmically divergent part traceless conserved currents we obtain the effective action teracting theory of“induced”) CHS action gauge of fields a freeing can scalar from be field the constructed [ action from of the a free effective complex (also scalar called field JHEP11(2018)167 r ], ⊕ 73 , (2) (2.12) (2.14) (2.11) (2.13) 72 so ) where . 0 2 d , ` 2 2 ` < − d ( V ) character of the ]). The interacting -lineton) describing is the distance from , d ` 71 )-weight, i.e. R (2 – d (or ( so 68 ` so and + 1] s ] r r . ) where we turn on, besides the Y = 2 = 2 ) is irreducible if only if . +1 d d i ∆; [ [ d , which is induced from the . Ω 2.14 ) ) β Y D e 0 0 , , , , ` ` ∆; = , | 2 r 2 2 i 0 − +2 s V 0 d d | ( ( ≈ > > corresponding to the angular momenta [ V V φ i r – 7 – 1 ` s , x − = r is described by the Verma module ]. Here ∆ is a real number corresponding to the β > s ` ) is an integral dominant ` − Y r 2 e > 2 is even the module ( )-module called Rac of order- − ··· d Rac )-modules relevant in CHS theory. = d , d > ··· q , d module 2 (2 th power of the wave operator as kinetic operator. Off-shell, , . . . , s > s (2 1 ` 8 so 2 s > so s )-weight. On-shell, such a scalar field obeying to the polywave 1 d > ( s := ( 1 so s Y . +1 ). The irreducible module obtained as a quotient of the Verma module ]. d d , the chemical potentials Ω are either all integers or all half-integers, corresponding to the spin, and ( 74 1 r of conformal weight so − (the conformal boundary of thermal AdS by a (generalized) Verma module φ β 1 7 − d by its maximal submodule will be denoted is the boundary-to-bulk propagator of Fronsdal field Φ , . . . , s S 1 We introduce first the s K × Y ]). As usual in a conformal field theory, a primary field (together with its descendants) 1 Suppose that we are interested in the quantum properties of CHS theory such as the ) module with lowest weight [∆; Strictly speaking, a generalized Verma module is theNotice algebraic that, dual strictly of speaking, the when infinite jet space at a point S 64 d ∆; 7 8 ( stands for the trivial of a primary field [ 0 equation, is described by the irreducible is the rank of V Rac. the conformal scalar field with a scalar field conformal weight, and where Let us start byof [ reviewing the relevantis modules described in the freeso CHS theory (see appendix F In order to computecharacters the of character the of individual the free CHS theory,2.2 we need to determine Relevant modules first the one-loop free energy. Then,free this CHS quantity theory. is The closely latteron related character to can the be viewedtemperature as a single-particle partition function upon the identification, the center to the boundarytheory of of the regularized CHS AdS fields spaceKilling (see enjoys tensors. e.g. a From [ the non-Abelian effectivethe action global maximal point symmetry of symmetry generated view, of thissymmetry by the CHS in the free symmetry AdS conformal scalar is nothing field. but Hence, it coincides with the type-A MHS where JHEP11(2018)167 s s − = in in 2 s s = 0 s = 4 J h J s (2.17) (2.15) (2.16) s S d s x h x h d d d unitarizable d R R . However, for d 9 = 0 is somewhat . 0 field with Dirichlet s ss is isomorphic to the s δ ). The precise group- tensors with conformal 0 = s . has a transparent group- 1) i 0 (2; s − tensor. This current corre- J 1) V ··· ) perspectives, the s s ]. s s − + h , 75 h s s 1; ( +1 ). The case ) = 2; ( h d s 0 − 1) 1; ( ) ∂ − 2.2 d d (2; FT (or shadow) field corresponds to − = s, s S d s, s, + + . s,s s s 2; ( + 0 3; ( C s V V V ≈ in the sense of [ D V s s off-shell solutions thereof. J J – 8 – ∼ = with respect to the inner product · = s defined as the quotient ∂ ) ). From the AdS J The ) s ) s, s s Weyl tensor 2.15 2; ( = 1, in which case it is simply called “Rac” (or scalar s 2; ( 2; ( for the off-shell FT field is irreducible and isomorphic to ` is the (non-local) prepotential of the conserved current: − for the higher-order Rac module gives precisely the − D d ) normalizable 4 d s s,s 2 + − contains symmetric and traceless rank- k d ;( + s s whose field-theoretical realization is a totally symmetric rank- = 6, in which case it corresponds to the usual Rac singleton as s ) 8 however, the order of the scalar singleton is greater than 1 and Let us now introduce the module describing the conserved spin- = s d − D ) ` D 2 whereas the module > 2 s 2; ( . In this case, the previous inner product becomes where d ;( − S − ) s d d s,s − k + s, s quotiented by the gauge symmetries ( + s 2 s s s,s . Let us stress that this description applies to any dimension ) corresponds to the Hilbert space of the massless spin- 2; ( V h S s,s with respect to the identity two-point function : x C k s s d 2.16 s = 1. When ) J J d · ⊂ V 4 ∂ R 2 ( − s d Fortunately, the equivalent field-theoretical description of the off-shell FT field ) More precisely, it is the “shadow operator” of s ≈ 1) 9 odd, the module = s ;( − d the module terms of the gauge-invariant spin- theoretical description ass a submodule of( the corresponding VermaJ module, i.e. remains somewhat elusive. Notice thatbut this is module rather is related not to ait the (quotient is of) contragredient the thereof. Verma dual module(s) Indeed,the of classical the path field-theoretical conserved terms integral. current current Equivalently, in CFT terms it is the algebraic dual of the conserved Off-shell Fradkin-Tseytlin field. the module tensor field degenerate: the off-shelltheoretical scalar description FT of field is this simply definition of the shadow field in terms of Verma modules divergence of such tensors.to As imposing a the consequence, conservationmodule modding law ( ( out this submoduleboundary is conditions, equivalent i.e. the The module weight ∆ = Conserved current. current i.e. a totally symmetric,sponds to traceless the and module divergenceless rank- singleton). The value part of theand CHS is theory unitary spectrum. only` for Notice that thetherefore this scalar field FT is non-unitary, field as is the on-shell absent FT in fields. This module is unitarizable for JHEP11(2018)167 has and given s (2.22) (2.20) (2.21) (2.19) (2.18) h ]). The 1) off-shell s,s ), i.e. the 57 − C ). At first s 2.4 2.4 ;( which has the s of the − ) ) s 1 FT field is in fact ) in the sense that ). Notice that there can be expressed as ;( s, s D s and conformal Killing 2.3 ) 2.18 2; ( − 2 +1 s, s D d . , on-shell S ) 2; ( . , x 1) 4 ( ) , on-shell Weyl tensor ) D 2 − ) 2)-module s d − d s ( s ) ;( s d, so Y ) 1) scalar singleton, is s ( s 2; ( ;( χ 4 ;( s so ) − s, s 2 s − − x s, s, 2 = Rac d − d − ) q, 1 2; ( U = ( 2 3; ( 0 + d ) ` V V s 0 2; P S U even, the module ∼ = ∆ − D = 0 is consistent with ( (2; – 9 – d ∼ = q d s V ) corresponds to the direct translation of the field- ( ∼ = ) 1) V , where the quotient by the irreducible submodule ) = For ) − x 2.18 s, s s,s s C ) = q, s, s ;( ( 2; ( 0 ) s The last module we shall introduce is the one correspond- Y 2; ( , ) may be interpreted as the left-hand-side of ( D corresponds to the off-shell FT field whereas the quotient − (2; (∆ 1 D D V ) 2.16 s χ D ;( corresponds to the (large) gauge parameters of FT field s Weyl tensor − has the interpretation of imposing the Bach equation ( implements the identities `ala Bianchi obeyed by the on-shell Weyl s 1) 2 ) − s S 1) s 2; ( , so the finite-dimensional irreducible implements the generalized Bianchi identities. ;( has the interpretation of pure gauge shadow fields, thus quotienting by this d s s, s, . Accordingly, the module corresponding to the − ) − d 1) s 3; ( 1 ;( + U s ). Again s s, s, V − 2.6 D 3; ( 2 2.3 Character ofThe the characters Fradkin-Tseytlin module ofcharacters the of modules the presented Verma module, in the previous section can be related to the U submodule corresponds to imposing theis conformal Killing a equation one-to-one ( correspondencetensors between on Killing M tensors onclear AdS bulk and boundary interpretations. ing to conformal Killing tensors, Here tensor. Notice thatthe the on-shell degenerate scalar case FT field, which is an order Conformal Killing tensors. another quotient using its Bernstein-Gel’fand-Gel’fand (BGG)latter sequence gives (see e.g. [ where in ( by glance, it might belowest weight curious 2. why While the the quotienttheoretical above ( definition quotient of module the is on-shell FT field, the module given by the following quotient: where the left-hand-side can be interpreted as the spin- D On-shell Fradkin-Tseytlin field. FT field is reducible, corresponding toInterestingly, the the possibility module of imposing ( theBach (local) Bach tensor equation. describing the spin- JHEP11(2018)167 ] ) ) s h 57 . 2.26 2.28 (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) A , is given, d 1)) P − s )-modules [ ), we obtain 1;( )) describing − , d d s , + (2 2.25 ;( s ( s 1)) so V . . − 1 s − χ ) ;( − 1 1)) of gauge parameters ) in ( s + q (2 − k , − )) − , V s ) on the right-hand-side. In (1 s in terms of Verma module → 2.27 q x . x )) D 2;( s ;( q ) and the function ). The heuristic behind ( )) χ − − s q, 1)) 1)) on-shell FT field — which had d 2;( ( − , d − ) s,s to the character of the conformal + + − − s 1 s s s Y d )(1 , 2.16 ;( (2 ( )) s k ) and ( (1 (2;( + 1;( V ∆ 1)) s under − so D s,s − ( V χ − − on-shell FT field. From the defini- d q x s d (1 d χ D ( − + ;( 2.26 D (2;( s P χ s V s − ) of ( χ D 1)) − χ d V − χ ( − (1 d and the Bach tensor (or conserved current) (1 χ s )) V ;( so 1) s )) r s − χ =1 s ;( Y − k s − ;( )) s − − s (1 – 10 – r − 2 V (2 )) 2;( s (2 − S χ ) = ( d − ;( S χ ) s d − x χ 1 q up to the characters of a few Verma modules. The , − + )) 1)). Inserting ( = 1 s s (2 ( − − ;( − V V )) s 1)) q ( χ χ s − − (1 s,s ) s (2 ;( Y ;( , = = s V s (2;( χ ) = − (∆ )) )) D − s s V x (1 + χ 1)) describing the gauge parameters. However, this removes too ;( χ D s 2;( (1 , by q, on-shell FT field module 1)) χ are given by − ( d − − D d − d s s (2 s )) + P ] using the BGG resolution of finite-dimensional S s ;( s ;( alone due to the presence of s ( ) follows directly from the definition ( χ s 64 2;( − D ) − Y (1 − χ d D 2.27 + (∆; (1 ) does not yet express the character s χ 1)). In more physical terms, the module ( V is the character of the subalgebra V D χ ) = − d χ ), we first find 2.28 ( s )) inert (by definition). For this reason one has to correct by adding the character of so Y ;( s s,s χ s 2.18 h Let us begin with the character of the spin- In this paper, we aim to find the character of the total linearized spectrum of CHS − (2;( D (1 χ Killing tensor module relation ( characters principle, we could further work outthe to modules get rid but it of will the turn latter out module to using be another useful relation to for do the opposite. In fact, the expression ( contains large gauge transformations — which areKilling physical tensor — associated module with the conformal which relates the spin- the character of thethe pure module gauge modes. Thismuch. can be Indeed, done when byleave the subtracting gauge the character parameters of D are equal to conformal Killing tensors, they The equation ( is that one should subtract from the character of the module where the off-shell FT field character character theory. For that, webeen need obtained first in the [ character— of then the sum spin- overkey all steps the of spins. the In derivation, but order interested to readers avoid can thetion find technicalities, ( more we details shall in present appendix only which is simply a consequence of the behaviour of both for even and odd Let us point out one important property of the above character, where JHEP11(2018)167 . 1 − q (2.29) (2.30) (2.35) (2.32) (2.33) (2.34) (2.31) ) enjoys: , . . ) 2 ) x x ) 2.28 , . q, x q, ( ) ) ( , the character )) q, ). This leads to x x )) d s ( s , , 1 1 2;( 2;( − − Rac 2.27 − − q q d χ d ( ( + + )) )) s s s s ( ( − D D 2;( 2;( , 2 χ χ ) and ( − − ) while the other is with d d . ) x − q + + ) =0 ∞ x s s ) s X q, ( ( x 2.24 , ( x 1 D D , − q, )) − and the scalar field with a kinetic χ 1 χ ( s ) q d d − ) which is relevant for type-A CHS ( ) ) x 2;( q ∞ , Rac ( − − − 1 1.2 d χ Rac )) − ) s + χ q s x ( +1 ( 2;( ) + ( d ) = ( D )) q, − x s ( x d χ 1) ]: + ) + = 0 as well, except that the first term of the 2;( q, q, − s 1)) ( ( ( =0 x − ∞ 78 s − s X d – 11 – D – s 1)) q, 1)) + ;( χ ( s = − s − 76 ( ) = ( s s 2 (even or odd). Finally for even − 2 D ;( 1)) x ;( s ) + , (1 s χ − ) 1 − s D x − x ;( − (1 χ (1 ). Interestingly, both of these terms are given by the =0 d > s ∞ q q, s X D V ( q, ( − ( χ =1 χ ∞ ) to derive the character of the CHS theory linearized spec- (1 s X + 1)) 2.29 Rac D − − Rac χ χ s ) ) = χ 2.31 ;( ) = x x s =0 ∞ 4 derivatives. Focusing first on the FT fields, we consider x − s X q, q, ( (1 ( − q, ) and ( ( )) D )) d s s ) = χ )) ;( ;( x s ) is the instance of the identity ( s 2.25 s,s − − q, ) = ( (2 ] (2 ) becomes (2;( x 2.31 V S )) on-shell FT field module coincides with that of the conformal Killing tensor D 64 χ χ q, χ s,s s ( 2.32 )) =0 (2;( ∞ s X s,s D χ (2;( D =0 ∞ s X χ The second line ofobeys the the above property formula vanishes because the character of the Rac singleton the series ( The first term inadjoint the module of right-hand-side the ofline CHS the using symmetry equality the algebra. is Flato-Fronsdal Re-expressing theorem nothing the [ but two the series in character the of second the trum. The theory containsoperator the containing FT fields of spin 1 to gravity. It applies toright-hand-side the is degenerate absent in case this case. 2.4 Character ofWe type-A shall on-shell use conformal the higher-spin relation ( gravity as follows fromcharacters ( of the conserved currentThe module, formula but ( one is with which is valid inof any the dimension spin- module up to just two additional terms: theory. This is thanks to the special property that the Verma module part of ( Note that the abovethe property relation is [ a simple consequence of ( naturally leads to an interesting and suggestive expression of the character of the full CHS JHEP11(2018)167 (2.37) (2.38) ) one (2.39) (2.36) 1). off-shell > Actually, 2.36 s | q | 10 -dimensional d 1 and ) separately, despite spins in the spectrum. < , /q | ) ), the spin- . ]. q 2 i | ) 79 , x x even ]. 2 ]) and this property au- 2.18 q q, ( ( 83 80 and in 1 – , )) s -anomaly of a q Rac )) 81 . a χ 2;( s ): the Casimir energy of free − d 2;( (see [ 1)) ) + + − 1.1 i s − d ). From ( ( s /q + D ;( 1 is ensured to vanish in any regular- s q, x s χ ( ( 1.1 − N 1 fields of all integer spin). D 2 → ) for type-A CHS gravity. 1)) (1 − ∈ χ (the right-hand-side). By construction, X d − s q +1 D s d S 1.1 + ;( χ = 12 CHS gravity with only s ), since the character of each Killing tensor i )) − × ) =1 ∞ (1 2 i s X 1.1 s,s R D -anomaly of CHS gravity is also guaranteed to , x χ a 2 = – 12 – (2;( 0 q N ( minimal D )) 2 X ∈ χ s s,s Rac Consequently, confident in the validity of ( χ = (2;( ) = − . The cancellation of the latter can be shown using the )) i 11 D 2 s ) imply the relation χ ;( +1 ) s d q, x i ( − =0 ∞ )) 2.35 s X (2 q, x ( S s,s ). In fact, the vanishing of the Casimir energy is ensured when χ (2;( Rac ) does not hold for 1.1 D χ χ 1.1 h N 2 1 2 -dimensional CHS gravity is just minus two times the free energy of ∈ ) involve divergent series which require some regularization. However, with Neumann boundary condition and the same with Dirichlet condi- X s d ) itself only assumed the validity of the Flato-Fronsdal theorem which sum of the characters of massless AdS +1 2.36 d 2.36 ) and the analogue of ( ]. Since the free energy with Neumann boundary condition is simply minus that alternating 2.31 46 There is another virtuous corollary of the relation ( Finally, we find that the character of all the on-shell fields in the free CHS theory Some convergence and regularization issues of the latter were addressed in [ The only assumption is that it holds for each sum of characters (in Let us illustrate why ( 12 11 10 -anomaly of the where the extra terminto on an the right-hand-side has no clear interpretationthe in fact this that context the (as corresponding it power decomposes series have distinct region of convergence ( In such case, the Flato-Fronsdal theorem involves a symmetric plethysm of the Rac module: Then ( 2.5 Character ofWe type-A can also off-shell derive conformal an off-shell higher-spinFT version gravity field of module the identity is ( related to the on-shell one by tion [ with Dirichlet condition (thea contribution of Killing tensorMHS module gravity in simply Euclidean vanishes), AdS method the of character integral representation of zeta function [ tomatically holds for the right-hand-sidemodule of obeys ( this property.vanish Actually, by the virtue of this propertyFT of the field character. coincides Indeed, with the Euclidean the AdS difference of the free energy of the associated massless field in important requirement of a sensible regularization but which isCHS usually theory not on guaranteed. theization Einstein consistent static with universe ( the partition function is invariant under the map both sides of ( the equality ( does not need anymight regularization. somehow reduce the issue ofhand-side) regularizing the to character the of the one CHSthe of spectrum corresponding the (the regularization left- higher-spin of algebra CHS theory would preserve higher-spin symmetries, an coincides with that of the global symmetry of CHS theory: This result can be understood as the equality ( JHEP11(2018)167 2, − d (2.40) (2.42) (2.41) + ). The s fields will = 2.41 + all . Accordingly, , s , one can make , d − )) and Chern-Simons s Free 4 CHS CFT theory theory 2;( = 2 Boundary − d − + s ( D χ Dirichlet MHS , = 4. In fact, the above off-shell field ⊕ =0 ∞ s X current Shadow d operator 2;0) Boundary d Conserved ) − d with conformal weight ∆ ( − s 2 V conserved currents, one is lead to the J χ s = = − + ( s s d + − + − ∆ 1)) 2 4 + ∆ ∆ − 2 − s s d ]), but the logic works for any dimension (see ;( ) have natural interpretations in terms of the (2; 0) is related to the on-shell one by s massless bulk field, two boundary conditions are – 13 – V with conformal weight ∆ − ] for spin-two at interacting level). Rac 90 s , (1 s χ 2.42 91 h even, this result can be viewed as D = 89 χ , d 46 =1 ∞ s (2; 0) = X (2;0) field Bulk S S = solution solution χ )) Normalizable s ], applied for all spin- ;( Non-normalizable s . The holographic dual of Dirichlet MHS theory is a free scalar (see e.g. [ 88 − – 1 3 (2 S 86 . List of relevant fields in Dirichlet vs Neumann MHS theories χ =0 ∞ s X theory (exotic) Dirichlet Neumann (standard) Bulk MHS Table 1 ]. Following the usual considerations on double-trace deformations and holo- ). Summing over the characters of these off-shell field modules, we arrive at even: Off-Shell CHS = Higher-spin Algebra 85 , with Dirichlet boundary conditions, corresponds to the last series in ( d 2.30 ] for any spin at free level and [ 84 +1 60 From a holographic perspective, another interpretation of this last result is possible, d graphic degeneracy [ conclusion that the holographicnario dual has of been Neumann extensively MHS discussedCHS theory for theory is the around CHS MHS M theorye.g. gravity. around [ This AdS sce- or the exotic (“Neumann”)tions boundary corresponding condition to which a allowsthe shadow non-normalizable field bulk bulk theory solu- withbe standard (respectively, referred exotic) to boundary asand condition Dirichlet compared for (respectively, in Neumann) table CFT MHS [ theory. They are summarized higher-spin algebra. purely in terms of bulkavailable: fields. either For a the spin- standard (“Dirichlet”)bulk solutions boundary corresponding condition to which a conserved allows current normalizable where the last term onAdS the right-hand-side, the linearized spectrumtwo of MHS terms gravity on around thehigher-spin right-hand-side algebra: of ( theyadjoint are module important while modules the of second the term latter. is the The so-called first twisted term adjoint is module the of the which holds for any dimension. For where the first termscalar on the becomes right-hand-side an is auxiliaryuse absent field of in in ( four dimensions. In odd dimension whereas the off-shell FT scalar JHEP11(2018)167 ] . s 92 h (2.45) (2.46) (2.47) (2.48) (2.43) (2.44) = 0 case). . . ) bilinear s 2 s ··· 2 s . . , x J σ ] 2 d N s , ∈ 1 d λ ] — where IGT 2 1 s ) 2 } , s , s 46 − Z h s h , j − { s , s s = 0 s N j σ λ σ 2 2 ( ,N s ) s s x h ··· x = 0 term is relevant and the h d , d d =0 ∞ − =0 , one gets s d s =1 X s ∞ s R i R σ } s ( (with i P 1 λ φ s =0 2 s { ∞ s λ ] + N [ ] + → ∞ J 2 N N N P =0 s ∈ ∈ ∈ ∞ s − = 2 the s s s λ s } } such that the new background field ] + d } P J s s N does not have any gauge symmetries. s s λ ) h 14 − ∈ { s j h { s ] s [ S h { } s N σ Λ is now again a Gaussian integral which s , − − ∈ λ s σ s e W which require a separate discussion (see e.g. [ } { φ σ ,N [ s ( = # d Λ i σ ··· s , { N φ [ W which might require a separate discussion which will x h − Λ d =1 D d d i W – 14 – } N R h e i N =1 − i Y φ D N σ e { " [ Λ − σ e D Λ Z S D Z σ e and dimension Z = Λ = ] D s Z ] = N = ] 2 Λ s ∈ )-singlet primary operators N N and dimension 2 s s Z ∈ ∈ these deformations are irrelevant in the infrared (IR). The } x j s N s s s d } x J j = 0 term is marginal while for } = s : d d { s i 13 [ h ] s d R { h N φ [ R s ∈ { UV 2 N Λ ] s λ , s ∈ N } N s λ 2 s ∈ } ,N s =0 h s − NW ∞ s } { λ [ ··· s e − , { = 4 the Λ j P N e s ∈ d s =1 λ N i } { e [ NW s } i λ N − { Λ ∈ φ = e s { } [ s complex scalar N λ NW ∈ { Λ s − } N e s λ NW This exotic type of holographic dualities where both sides can be gravity theories { Except possibly for very low spin Except possibly for low spin − through Gaussian integrals as follows: has the bare scaling dimension of a conserved current, S e 13 14 = 1 is marginal. s s for detailed discussion of theMore double-trace deformation precisely, and for its holographics interpretation in the be avoided here for the sake of simplicity. Considering the vicinity of the UV fixed point where One may then perform thej field redefinition The integration over thecan dynamical now scalar be field performed leading to σ where we neglected an infinite prefactor. Note that corresponding effective action is defined as The standard Hubbard-Stratonovich trick corresponds to the introduction of auxiliary fields in the These deformations explicitlyNotice break that all essentially gauge all symmetries of the background fields (though of Einstein vsstands Weyl for type) induced has gaugecorrespondence. been theory This denoted type — of “AdS/IGT” inone, holographic in so duality order [ is let to somewhat us less distinguishtrace expand familiar deformations it a and for from little is all a bit U( standard subtle and AdS/CFT present some details. One starts by adding double- JHEP11(2018)167 and (2.50) (2.49) . . ]. N ∈ ] for a manifestly s } s 30 h spacetime of even di- { ) module in one-to-one [ +1 , d log d (2 W Dirichlet MHS so ⊕ due to the disappearance of the ), the Hubbard-Stratonovitch field → ∞ 2.48 s theories λ shadow field (see e.g. [ ` s – 15 – . In ( 2 s theories. Neumann MHS = Higher-spin Algebra x j ` d ⊕ d R and type-B ] is the Legendre transform of s (i.e. semiclassical) limit one has that the logarithmically 2 ` N λ ∈ s N =0 } ∞ s s odd leads after summation over all spins to a relation between ) can be rephrased purely in bulk terms as follows: and type-B j (consistently with its bare scaling dimension) in order to stress { P ` d [ s conformal higher-spin symmetries (rigid symmetries in the former ] stands for the UV-divergent part of the functional h 2.42 UV N Λ Focusing on the logarithmically divergent piece, the left-hand-side ] + ∈ s N ) for W } ∈ imposing any boundary condition (i.e. considering both normalizable s s ) would require a careful discussion of the the ghost contribution in the measure. This 15 j } { s 2.30 odd: Dirichlet MHS [ j 2.48 unbroken s d UV Λ λ { even: Neumann MHS = Higher-spin Algebra [ without W N ) can be interpreted as the generating functional of the correlators in CHS gravity. d ∈ MHS fields with Neumann boundary condition is an s } s s As a side remark, one may observe that, the opposite sign in the last term on the right- The holographic dual to a Legendre transform on the boundary is a change of boundary 2.48 Therefore ( λ has been denoted { Λ 15 s coincides with the character of thesection, adjoint module we provide of more the non-trivial type-A higher-spin evidencesthe algebra. of result this In to intriguing this observation the by type-A generalizing will not be performed here because we only intend to sketch the logic of the proof. dynamical degrees of freedom reorganize into its asymptotic symmetries. 3 Extensions to type-A In the previous section, we have shown that the character of the type-A CHS theory This relation suggests thatmension the linearized MHSnon-normalizable theory solutions) on might AdS also be somewhat “topological” in the sense that its points with case, gauge symmetries in the latter) since all spinshand-side are of on ( the samecharacters footing. which one can rewrite as In other words, ourall character extra computation dynamical suggests degrees thatmodified of asymptotic from freedom charges Dirichlet in account to MHSDirichlet for Neumann theory and when ones. Neumann all Let boundary MHS us conditions theories stress are are that (respectively, both holographic infrared duals and to ultraviolet) fixed conformal and gauge invariantspectra field-theoretical description). of Consequently, off-shell theTherefore CHS linearized the theory result ( and Neumann MHS are in one-to-one correspondence. In particular, indivergent the piece of large- conditions from Dirichlet to Neumannspin- on the bulk fields.correspondence In with fact, the the solution module space of of the a spin- W σ that gauge symmetry isquadratic restored term. in thein limit ( where JHEP11(2018)167 in 2) FT ` − (3.7) (3.4) (3.5) (3.1) (3.2) (3.3) s s conformal t a rank-( 2 − s . σ , via a set of currents +1 , parameterizes these t ) s ) (3.6) t − ( s like the usual FT field, ) , t 6 h ( s,s ) ) + 1 t s t t ( s ( s C t h , we can find currents of all h 4 , . . . , ` − 6 ` 2 2 − 2 s 2 − d , dimensions, the action for the d which becomes local after mul- . + d 0 , t ), and it has the conformal weight 16 = 1 2 − tensor +1 1) ≈ s t − k − s ) 3.5 ( s − t ( ) ,d s ∼ = 0 defines now the depth- k . t T ( s,s J t η σ (1 2 ) (so that the usual FT field corresponds t t s T ( s ) so to determine the action uniquely. · + − P ]. These currents take the form h ) , ∂ x C t ) t d t ( s ( d +1 ( s − t ξ,σ h 94 d s δ , + − ··· the Minkowski metric and ξ depth d s = 2 . In even t s x h +1 η ] + M t – 16 – d ∂ FT fields can be obtained as an effective action, 93 , , T Z − = d +1 φ t 31 t t s 0 = s T d k − 31 ∂ P ) ) +1 s,k t − M t ≈ t ( ( s s,s ` Z ) +1 1 [ − ∆ C h t t s ( s theory − s − J ξ,σ 1) ] = takes value inside the range 1 ` ` · δ ) s,s T t − 2 ¯ t ( s φ ∂ η P -ple transversality and tracelessness does not fix the projector uniquely. t h -ple divergenceless: . The condition [ t = = ) t ( ] = ( +1 1) ) ,..., t t − +1 3 ( s FT − t k , s h S − [ (2 s ) symmetric tensor, ) ) t t t J ( s,s ( = 1 − FT field is then given by C -ple transverse and traceless projector t FT = 1 case. We replace the free scalar action by its analog of order 2 t s s S t is the a rank-( T t t and spin- s T − P t s ξ = 1). As in the usual FT field case, we can define a gauge-invariant field-strength, An interacting theory of the depth- which are traceless and Note that the condition of the t ) t 16 ( s where the “. . . ” stands for additional terms ensuring ( We need to impose the locality condition on One can show that forintegers a spin given with free scalar action with a fixed similarly to the the derivatives and coupleJ the system to the higher-spin sources where tiplying by the factor Killing tensors. After integrating by part, the action takes the form of which also has conformal weightdepth- ∆ with symmetric tensor. The integer class of fields andto will be referred tothat as is, the a Weyl-like tensor as Let us introduce another classfield. of conformal They gauge are fields which describedbut are they by cousins have totally-symmetric of weaker the rank- gauge spin- symmetry [ 3.1 Field theory of type-A JHEP11(2018)167 . ) is 2 d s and 3.5 (3.8) (3.9) s (3.10) (3.11) , ) , all these x 4 d , ( ) ) is the local and spin d ! ( correspond to 6 . δ k 1) ) ` ` 2 − 3.10 3 k . This interacting h 6 − ) (2 s ( ` . The latter bound ··· ( h d log k 2 derivatives but do s O 1) 4 + + − 6 +2 − . From the CFT point d k k s κ W d ] + 2 4 d (2 s 2 1) +2 − 6 = s J − 2 d 6 . k ) + k + t ) ` ` (2 s } s =1 ` ( CHS k X h 1) − [ S − : s 1) s ` : =0 ∞ η − 1) s 1) and X +2 k κ − k (2 k FT + − (2 s (2 S s imposes ` s,k defining those fields as their depth. In φ J = 0 case gives the lowest bound, and J ∆ ` ` ` t ` =1 2 ( scalar singleton. When , and hence s ρ k X 2 ` 2 ¯ 6 0 − φ d =0 ∞ ∼ s s
X + 1. Although they do not enjoy any gauge + ,..., x One can note here that the contact terms are 6 s = d . The s – 17 – . · · · −→ ` d +1) > k } d 6 + (2 t s t M J , we obtain an effective action. Again, the logarith- ,...,` Z + 2 φ , k 1 ]. We will refer to this class of fields as “special FT” ) Γ(∆)] = 4 , 2 with − 1, 1) ∈{ 95 2 s d − } t − d η , k k − +2 N (2 + ` s s , namely in [ ,...,` − ∈ 2 t 2 do not satisfy any (partial-)conservation condition since ( 2 J s | s , { 1 } x | -derivative scalar field action in the background of higher-spin − 1) ∈{ ` ), the fields with 0 Γ(∆ − ,..., k , k d t > s 3 i ∝ (2 s special N − , can be regarded as an action of interacting higher-depth FT fields up h 3.10 ∈ . s ) { 2∆ (0) ` s } ( log = 1 ) 1) [2 with 4 1) ` +2 ( − t / log − W ) d k t 2 d k ( s (2 s W (2 s π J J h ) { k > = ; x , it has the structure ( φ ∆ 4 d 1) ρ − S k 6 Starting from the 2 (2 s ` J h where absent for hence the condition thatof this view, be the not bound smaller impliesThis that than bound there can is actually no be operatorsphenomenon relaxed discussed with without below. conformal encountering weights The lower an origincontact than inconsistency of terms due the hidden to quadratic in a Lagrangian the subtle in two ( point functions of this class. The freenot Lagrangians have of any these gauge symmetry. fields Anumber still trivial of has but derivatives 2( important of restriction theirdependent to free these upper Lagrangian fields bound cannot is for that beis the negative. irrelevant This when gives it a dimension is not smaller than theory contains not onlyfields, the referred higher-depth to FT as for fields the but sake also of other uniformityconformal non-gauge in weight the conformal ∆ terminology. =symmetry, They 1 we correspond will to keep referring thethe to fields the quadratic of parameter part spin- ( The functional to the introduction of a dimensionless coupling constant and integrating out themically scalar divergent field part of thefor effective action is a local functional of higher-spin fields, and fields of depths is not defined infor such the case. space Still, of theseoperators operators tensors are bilinear can primary, and in be the thespanned used space by order- as of the part operators basis of with the dimension basis operators The tensors JHEP11(2018)167 ) . can . scalar 3.13 d 1) (3.14) (3.15) (3.16) (3.13) (3.12) ` − 0 k s (2 are in fact with ( HS symme- on the CFT MHS theory s ` 1) ` and partially- 4 − +2 k d and Φ d (2 s 1) ), and the current ˜ h . > − ) 0 k 1 FT fields in M is the corresponding x k s (2 , ( t ) . extension ) and t Interestingly, the cross 4 d ( , the operator spectrum d 1) ( 4 d − 1) δ K 0 17 − k with s 0 ` > η k (2 s 1) ` > (2 s h 2 d − h 0 and we dropped numerical co- ρ 1) : k . will lead to the effective action − (2 s 0 k 0 1) +1 . h k d (2 s − ∼ 1) ] that the AdS counterpart of this k ˜ − h 1) k 6 (2 s − 96 ˜ 0 (2 + J s k · · · −→ ˜ J (2 1) s ). As a consequence, we can find a new + − J k k (2 s 3.8 theories (CHS gravity in M , k + − s 0 ˜ h with the same spin but with the respectively d | k η ` 2 d ) in higher-spin background, where all currents x 1) | – 18 – − + ] + (regular or polynomially divergent terms) 0 does not vanish but gives ) 0 . Hence, both of , in other words, ··· 3.9 k t ∝ s ( s d (2 1) s i ∝ h ≈ + 1) − [ = J = k − ) partially-massless field and k 0 t 1) k 2 0 ( (2 (0) s k t − (2 ˜ s − J k 1) FT s,k 2 d J and + − (2 s ) are replaced by k + ˜ h k 1) s (2 ) have the same global symmetries: the type-A s RS − and ˜ are both primary and descendent. Because of the degener- + ∆ J k +1 3.13 ) (2 1) s d 1) x s,k − J 1) − 0 partially-massless field in AdS ( k k − 1) and depth- t k (2 s (2 s ] = log − (2 s which is neither primary nor descendent. ) 0 J s J t k ˜ h ( s (2 1) s h − ) J t k h ( (2 , we face an interesting phenomenon referred to as s K ˜ 4 d J are the source fields of the operators = ) 1) t ` > ( s is the spin- . Analogously to the usual FT/massless case, the depth- − ]. Let us briefly review this extension hereafter. For , none of these currents are (partially-)conserved (see figure k ), we loose one basis operator from ( ) 2 d [Φ t (2 s +1 s ( ) 93 with the condition ( t d ˜ h ( containing the quadratic terms 3.14 1) When This higher-depth CHS theory appears from the on-shell action of type-A ` < PM ) − Notice that this phenomenon is a consequence of the fact that the tensor product of two order- ` k S ( log 17 (2 s Lagrange multipliers enforcing are absent in theextension on-shell phenomenon spectrum. is the It mass is term shown mixing in between [ the fieldssingletons Φ contains reducible (though indecomposable) representations for where efficients in each terms. This shows that the FT fields This implies that the scalarJ field action ( W Hence, the operators acy ( basis operator two point function of For operator with higher conformal dimensions becomes a descendent of the other: contains pairs of currents dual conformal weights, ∆ boundary-to-bulk propagator. Bothmassless type-A HS gravity in AdS try algebra generated by higher-depth conformal Killing tensors. side in [ in AdS be related to the depth- Here, Φ JHEP11(2018)167 = 1 . The (3.17) ` 2 d having , we can ]. 1) 2 d − 0 ` < k 98 , s (2 scalar CFT is s 6 97 ` J [ 4 d , s 1 t − − s . s 1) − 0 ]. We postpone the analysis k ) , of the order- (2 s 93 , s J = 2 case is the familiar example. ∆ 1) 2 d ` − 2 d 0 2 − k d − (2 )) of the on- and off-shell FT fields of s d s J ( 1) 1) − − 0 s, s,k k k , whereas the right one with g = 1 and (2 s (2 4 d 0 contains a conformally-invariant subspace. s − 0 ˜ ` J J t s,k ≈ X ]), and ) have conformal weights lower than -ple conserved current of spin- – 19 – ` < φ t + 31 ` (1 + a CHS theory are analogous to those of the ) two-row Young diagram 3.13 S φ may arise. From the AdS point of view, the double- ` ` , d s )) of 1) s (2 − a k ¯ so φ with ( (2 1; ( s )) and theory J t 1) − )) of the conformal Killing tensor which is a finite-dimensional ` x − t 0 − d d k ) free CFT with the double trace deformations of such operators: − d (2 s s + d N (introduced in [ J s, s t ;( M ( t s , Z − . The zero mode of the − (2 s 2 d ( (1 D − D D ` , the solution space of ∆ 2 2 d ` 2 d 2 − d − > and depth- d ` s . The spectrum of bilinear operators, labeled by (∆ . The left diagram is the case with ` the modules spin- the module irrep corresponding to the the module When Since the operators • • • trace deformation corresponds toIt changing will the be boundary interesting conditions toof of clarify double-trace the how deformation/changing the relevant boundary “extension” fields. phenomenon conditions. will affect the mechanism 3.2 Character ofThe type-A modules relevant totheory: the they are type-A Hence, we are lead topart. consider either In the fact, finite-dimensionalmaximum the quotient order quotient part, space or the correspondsThe subspace to CFT the based space on ofof the harmonic CHS quotient theory polynomials part in with has this been case studied to in a [ future work. the same conformal weight as think of deforming the U( This will lead to an interacting CFT in IR, where a new class of operators without conservation condition and with (partial-)conservation condition.lines is The number of the dotted operators in the shaded region do not give rise to on-shell FT fields. Figure 1 depicted. The black solid circles and the blue empty circles designate respectively the operators JHEP11(2018)167 . ) + 2, x ] (3.21) (3.22) (3.23) (3.18) (3.19) (3.20) 2 d q, )) with ( 1. 99 s , − )) s − ;( t 31 s 1;( . . , > − − ) i ) t ) , s t − x x ), and they are d , and x , . . . , t # q, q, + ) 1 )) , d ( s q, ( ) , s ( x ( d )) ,... x (2 (1 + D s )) P 1;( q, s χ D ( q, = 0 so − 1;( ( ) + 1 d )) 1;( s − − t )) x + d − s t − ) ( d + s ) t, t − ) t t x + 1;( s ;( d , t , ( − ( s − − = 1 ) s s − d D − ( so ( s − CHS theory. Here, the module s x ( + χ V (1 χ q t ` V t ( t χ q, D − χ q ( = s )) ∞ χ ` ( s s X − t − V Flato-Fronsdal theorem [ − ) = 1;( + ∞ ) χ ) ` Rac 2 s x X . The modules − x , x t χ )) d − ( , 1 ) s − ,... 1 ) 1 ) 3 − d x +1 d , − 1 and spin 1;( − ( ) d q x ` + s X ( q 2 − s q, so ( =1 ( ( − 1) d ( q, t )) χ ` operator without any conservation condition. ( s D ` )) + − t s 2 χ )) s 1;( − s 1 Rac – 20 – 1;( ) = s − ;( ( − χ s d − t x ) for even V ) + d ) = ( − + − t ,..., t χ x q, + d x t 3 ( − − 1 , + , s − 2.24 q, s 1 =0 − 2 (1+ ( s ( t s q X ( − V V )) +1)) q " V ) t CHS theory consists of χ χ t = 1 ( χ x ` − − ` t ,... 1 ) = h s , = 3 1 , ;( − 1 x s,s ` s Rac ) = − X =0 − 2 − =1 t q s q, χ X x ), t (2;( ( ( (1 ` D q, )) ,... 1 D = s ( 3 χ , − χ 2.31 2 ` Rac )) t X 1;( 2 s =1 χ = ∞ − ;( t ` s . What is important for our purpose is that these characters satisfy an X t s ) = − − )) corresponds to a spin- + − A ,... t d 1 x Rac s arise from the “non-standard” BGG sequence of 3 , + − χ ` s q, X ( (1+ 2 ( 1; ( =1 t D D ,... χ − χ +1)) d t − + 1 + s,s t the FT fields with depth the special FT fields with depth in the same range but spin s, s − (2;( • • s D = ( χ Here, we used againt the property ( associated with the special FT fields. This module exists in fact only if and in the thirdFT fields: line we find the character corresponding to the module of the special Again the second line vanishes due to the property We first sum the characters over the higher-depth FT field modules and obtain we can handily compute theV character of the entire type-A The field content of the type-A Using the above properties together with the type-A whereas the characters of thegiven other in modules appendix are more involvedanalogous and relation their to explicit ( forms are The character of the partially-conserved current module is simple: JHEP11(2018)167 ` s . > , i.e. t )) ` t (3.27) (3.28) (3.25) (3.26) (3.24) − s , ;( is related s )) s − t . In the end, )) with (1 − 6 , s 2 d s D ) t ;( . ;( s χ x 4 d , it becomes s t − 6 2 d ` < = q, ∞ (1 − s ( X if D t − ` < )) χ ) contains terms with s ` ,... t 1 spinor singleton which , 3 t . , − 1;( ` + ` = )) 6 ∞ X (1 + 2 )) s s − X =1 3.21 s s d t V +1 is smaller enough than the 1;( + 2 d ,... 1 1;( t − 3 = ` − , ) holds for any value of − t − ). − t ` t s X − 2 ( =1 )) = − d cases with t d 1.1 6 D s + FT field with 1 2.42 ` + s χ s ;( = ( s +1)) t t ( s − D # V 6 − χ − χ s,s t + ) = +1)) + + 2 MHS theory in an analogous way to the t x (2;( 2 d )) − ` D s q, (1 + +1)) ( ;( − s,s χ t s S t t )) − − s = t (2;( ∞ s s,s X – 21 – ) in the type-A D 1;( − χ (1+ where the third line of ( + with (2;( d t 1.1 D 4 d D + )) = )) χ t ∞ s s s χ X − ;( > ;( s s = CHS theory based on the order- s ( + = ` − ` V − t )) t )) )) s χ s s theory ;( ;( (1+ ;( s (1+ − s cases. ` s , and we obtain D − + 1, the term simply vanishes. For D ) − t − 4 d t 2 d χ t 2 d χ x } , ` (1+ 1 (1+ − (1+ + S ` < − ` < D S t χ ), one can check that the relation ( q +3 χ χ ( , and therefore confirms once again ( 2 d ≤ = 1 . It is related to the type-B )) − The character of the off-shell depth- 1 2 d t 4 d s s =0 − ` , 3.28 t − s X t 0 X 1;( " { − ` < d ,... and we find 1 3 + so that we do not encounter any subtle issue analogous to what we found in , 6 − t theory. 4 d 2 d ` =max X ) and ( 2 − s 4 d d ` =1 s t + 1. For ( case. Here, we focus on the case where the value of V 2 ` < d ` < 3.27 ,... 1 χ ` 3 , − 6 − ` X Let us consider now the case t 2 4 d =1 t 6 Finally, let us consider the type-B will be denoted Di type-A dimension the type-A Using ( both of 3.3 Character of type-B whereas that of the specialsatisfy FT field modules Going off-shell. to that of the on-shell one through The left-hand-side of thetheory equality for is precisely the linearized on-shell spectrum of type-A and cancels the characters for which confirms again the observation ( s condition is satisfied if which is the same condition that the special FT field action has positive derivatives. This JHEP11(2018)167 . ` ) )) , 1 # − 3.19 r − (3.31) (3.32) (3.33) (3.30) ) hold 1 , s, ) ) 1 + 1) 1;( . 2.42 x − t , − )) t m q, ) − ( m − 1 x . d 1 , ] )) + s, q, m s s, s 78 ) and ( ( 1 ( 1 1;( + 1 t )) s, D − t − 2; ( t 1.1 m χ − 1;( s 1 − − s − d s, D + t + t − s 1;( = 0 counterparts ( )) d ( s, s = ∞ 1 − s + t M D − s r + − ( m χ )) . The vector space of the 1; ( d 1 ⊕ m D ), and use the Flato-Fronsdal m s, + =1 χ ∞ )-modules as [ s 1 D ) 0 s X ( 1;( s V − +1 s, , d m, 1 3.29 − t δ t ;( ∞ =0 χ ) − t M = ] for the field-theoretical approach): (2 − r − X m insertion. Note that the character , the generalized Verma module is s x 1; ( ∞ d − 0 , so M + ⊕ ) +1 + 1 ) t − 1 100 s s m, +1 − ( x = t ) − δ d q , s ` D − ) ( γ 2 1 m , whose Killing tensors form the type-B − 1 χ − t, x )) 1 1 − 0 =0 − δ d MHS theory, which in turn is given by the q m r M m q, ( s, m, 1 through ` D ( − δ + )) ) s, CHS theory ( + 1 ;( )) d 1 1 t s m ` – 22 – − =1 − 1;( m ( − 1 ) together with the ] (see also [ t s (3.29) ` M 2(2 1 + − s, ) V 2 t t, " 1 99 t, s 0 − ) − 1;( δ ; − + 1 3.33 d s =1 1 CHS theory, ` s < t t − t − X t ;( 2 t + − − ` s s ` )) = 1 − ( 2 − + d > m t, D (2 ) + (1 1 (1 + δ 1 D χ s 0 s D ( − D =1 1; ` s, ) and ( − V t t χ + M =1 2 − 2 s χ M t 1; ( 1) − 2(2 ' 3.32 d − ) = ( 1) 1 ` − 1 ) = − D + 1. If the latter condition is not satisfied, then we get the special x − =1 )) with t 2( − =1 ` x ` t t M χ r M M 2 m − m should be understood as q, 2( − 2 q, ( theory. = 1 > ( ⊕ ⊕ t d )) )) ` 1) )) 1 s, s 1 − 1) − − + m Algebra ` r 1 m − s 1; ( 1 ` 2( ` 1 s, X ( , ) to simplify the series. In the end, we find the relations ( s, − 2( ;( − t D = +1 1 and 1;( t CHS = t + t s − ` − 3.30 t ) over the field content of type-B > = − − higher-spin algebra can be decomposed into − s,s 2 Type-B (1 d d ` m ` D + 3.23 s (1;( Di χ + ( The field content of the type-B D χ D s Type-B χ ( χ We sum the charactersand ( ( theorem ( also for the type-B where FT field module: The character of the conformalof Killing the tensors corresponding appearing FT above field is module related in to M the character irreducible: type-B possesses the pseudo2 higher-spin currents with This theory contains infinitely manyD higher-spin gauge fields corresponding tohigher-spin the modules algebra. Note that for The universal factor 2 arises because we consider the parity-invariant Di module which can be read offcorresponding from Flato-Fronsdal theorem [ that of the the type-B JHEP11(2018)167 . ) 2 )— x (4.1) ↔ 3.33 . Hence, q 1 − q ) + ( 1 → ) and ( x q . For this reason, ` ↔ ) — is in fact very 3.23 , only the character q 1 − 3.32 q + ( -dimensional conformal 2 d → ) ) for the most general case. q q 1 and type-B − 2 ` x A.40 ) and in ( − only after an analytic continuation, 3.19 . For instance, for the type-A CHS )(1 ]. 1 q x − 2 q with different boundary conditions. The x 102 ) 2 − and +1 → q d q q − )(1 – 23 – q (1 1 counterpart theory under the flip ) character of CHS gravity spectrum and showed q − 1 x , d +1 d theory is given by a series which is convergent inside (2 − theory is symmetric under so +1 )(1 +1 d , plus — if the field has gauge symmetries — the character q d 1 ). Note that this relation is valid also for non-gauge fields ), which provides a simple link to its associated conformal 1 − x q 1.2 − A.44 ] and type-AZ theory [ (1 , we established the relation for partially-massless fields of any depth 101 ] for more discussions). After these regularizations, we obtain the ) — whose concrete versions for partially-massless fields of symmetric A − 79 1.2 replaced by 1. It shows the symmetry ) = 2 q < ), then about the summation. , x = 4 we find 1 | d q 1.2 | q, x ( 4) , Let us turn to the issue of the summation over spins or field contents. Since we have The relation ( (2 so CHS distributions (see [ character of CHS gravitytheory as in a function of χ the disk and hence thecharacter cancellation of of the higher-spin these symmetriesof is terms given convergence. already by a involves Nevertheless, serieshaving a we which does different can regularization. not regions still have of any evaluate Next, region convergence, this the and series this dividing procedure it makes into sense the only pieces in terms of for the type-AB case [ just equated, as series,not the elaborate character on of whetherFirst, this CHS series the gravity is character and convergent of that or the of not. AdS its In symmetries, fact, there we are did several issues. of the Killing tensor.fields Therefore, the is character always of given adifference by theory of the the composed character character of of multiple of thewhen conformal AdS its the global character symmetries of the andof AdS the the global rest symmetries whichour survives is observation as the extends is neither the to case the for type-C type-A case nor to the minimal theories, but holds Another way to statefield, this irrespective property of is whether thatthe it the characters has character of gauge of thecharacter any symmetry associated for or field the not, in field with is AdS condition Neumann given but condition by is the the same difference as of the character with Dirichlet and hook-symmetry types aregeneral. given In respectively appendix in ( and any symmetry: see— e.g. ( for totally-symmetricwhere and only hook-symmetry the type Killing fields tensor we contribution have is ( absent: see ( Killing tensor, the collection of which spanis the that, CHS symmetry in algebra. the Ourcancel key character observation out of after each therelation gauge ( field, summation. the Below, contributions besides let the us Killing make tensor two remarks: first about the key In this paper, wethat calculated it coincides the with thatalgebra. of its The global evaluation symmetry ofeach algebra, the field namely appearing full (conformal) in character the higher-spin spectrum requiresfield of the CHS satisfies gravity. summation the The of character relation of the each ( characters conformal gauge of 4 Discussion JHEP11(2018)167 1 → (4.4) (4.5) (4.6) (4.7) (4.2) (4.3) x 1 of the last ) 2 → regularization is , x , we can attempt 1 . 2 ) 1 3 1 , i.e. 2 − , x = d = q, x 1 x (4) character, ( . x +
. x s )) ) ) so ( ) s x x 1 − − 2 4) x q, e , − +2;( x 0 but not the simultaneous ( ) e s . ) (2 1 ( 0 , − ) . ) so V x -plane, so the contour integral → 2 e ( ) ) χ q 1 + 2 s,s 2 4) 2 , (3) in the q − 0 expansion leads to the partition , x (2 (2;( so s − 6 3 1 )( (1 + + 6 ) so D χ 2 + 11 ) consecutively or simultaneously to 1) − x ) q q → 2 d χ 1 or q q 1 2 2 q, x ) − − ( x 3 − , x ) + ], a closer investigation reveals that the − 2 1 e 2 → − − d 1)) x 64 (1 e ]: x (1 + 4 − + 6 (1 1 ( and s s q, x 2 x ( ( 64 1 q ( ) (11 + 26 -anomaly coefficient and the Casimir energy of = 4) 0 x − , 2 ) a +3;( – 24 – 2 ) e (3) s q (2 1 2 ( ) with a damping parameter − s,s so s 4) so V 1 + , − =0 ∞ χ s s X χ ( to 0 and then extracting the finite part in the (2 (2;( = ) = so V q − ) = x ) = χ +2 -anomaly of CHS gravity can be also shown using the ( e q 2 a d π i ) ( 2 x , x CHS ( 1 ) = CHS C ). Z 2 x I (3) ( Z 0 , x so s ) 2.4 1 χ (4) s,s ) := so ( ], so the spin-dependent part is simply the q =0 q, x ∞ χ 0 ( s X ( ) ]. As already detailed in [ )) CHS s,s s, s ) is singular, so we would need to subtract the finite part in the limit. 4) 80 Z , , (2 (2;( 4.1 38 so D ), whereas setting first χ 4.2 is a contour encircling the origin of the complex = 1 over all spins. For a concrete understanding, let us consider 2 C x = to 1 and then extracting the finite part in the 1 It is interestingenergy. to note The that cancellationcharacters the of (see the above subsection partition function also gives vanishing Casimir The above function islimit. regular And in depending either onx limit the evaluation order, wefunction indeed ( obtain different results.expansion leads Setting to Therefore, we see thatx the issue is in fact how to sum the above character evaluated at because the second and thirdsum terms over in the the spins. above expressionlowest The lead weight first to [2; ( term convergent series is as the we character of the generalized Verma module with CHS gravity [ subtleties lie in the first term of the on-shell FT field character, Here picks up the finite partintroduced to of ensure the the integrand. vanishing The of shift the The above was obtainedindividual by characters summing evaluated in the partition functions of each FT field (i.e. their to get the CHStwo partition terms function of from ( However, this this character. procedure is The somewhatMoreover, limit the ambiguous simplest as trials it (e.g. dependsdo sending not on easily how reproduce we the take this result found limit. in [ Since the one-loop partition function is the character evaluated at JHEP11(2018)167 α 1) we − (4.9) (4.8) 20 (4.10) , d (2 ], we can so 82 , , ]. Notice that 81 = 4 dimensions 2 β d 2 45 , and the variable β using the above char- sinh 40 2 α , 4 2 2 α 39 , . 2 ) factorize into a product of sin 6 0. One can then extract the d sin 4) [ π − (7) 2 β, α α ζ → ) ( > 64 0 expansion which leads to the α )) α s d ) and its derivatives (with respect − cos ;( → t 4 2 β cos 4.8 − π α 2 (5) 1). For instance, in 3) − , +2 ζ 48 s β (2 ( − sinh so D − Taking a definite boundary condition χ , d 2 ) ) in its π (2 γ 19 (3) + 8 (cosh so s ζ – 25 – 4.8 ( 192 − − . Using the method introduced in [ e ) = = iα s e ) onto =1 4 t β, α X ( , d = (1) AdS )) =1 ∞ s (2 Γ s x X ;( t so 0 expansion and compute the one-loop free energy using the − 0, which is singular here (contrary to MHS theories and their partially-massless fields with all possible depths (together with massive fields in dimensions 3) Fradkin-Tseytlin fields around AdS and , +2 → s s s (2 s ( → β so D , the on-shell Fradkin-Tseytlin module branches into the − α e χ C s = 3) character of this theory reads =1 t X , ) in q α (2 =1 ∞ s X so 6, the branching rule will also contain an infinite number of massive fields with arbitrarily Consequently, the CHS gravity around an AdS background can be viewed theory can be obtained by simply branching the representation carried by the > = 4, the partially-massless fields with Dirichlet and Neumann boundary conditions correspond interacting theory of massless, partially-massless and massive (with “the mass d d 18 d with all integer spins and depths. 4 so that the resulting characterfinite has part a in finite the limit in To extract only theone-loop finite free part energy of ( To introduce a damping factor in the summation of the character, ]. From a group-theoretical perspective, the factorization of the kinetic operator of local Let us conclude by a few comments on CHS theories around an AdS background. The A noteworthy subtlety here is that here appear both ofWhen the modules corresponding to the solutions with When • • 41 18 19 20 Dirichlet and Neumann boundary conditions. high integer spin. to isomorphic modules. compute the one-loop freeacter. energy To of do this so, theory oneto in needs the Euclidean to variable AdS evaluate thepartially-massless character cousins). ( As ascheme to consequence, cure this one additional has source to of divergence. introduce We a can easily new think regularization of two possibilities: where we have writtendefined the through character in terms of the temperature as an AdS CHS gravity spectrum from the type-A CHS theory spectrumin branches AdS onto the directfind sum that of the partially-massless fields Fradkin-Tseytlin fields is relateddetailed to in appendix the branchingmodules rule of which the correspond correspondingoperator. to module: the kernels as ofas a each factor ofsquared” smaller the than Fradkin-Tseytlin that of kinetic massless fields) fields of arbitrary spins. Such a field content kinetic operator of spin- kinetic operators of spin- a finite number ofthis property spin- was recentlyin generalized [ to the case of higher-depth Fradkin-Tseytlin fields JHEP11(2018)167 ]. 37 , leads (4.11) 36 +1 d . 4 ] the non-local free scalar CFT ]: MHS gravity in AdS π (5) 65 103 ζ 384 − 2 π as in the case of MHS or CHS theory in (3) 1 2 ζ 7680 = ]. The second question is what might be the – 26 – − γ 103 = 4 )-modules, as well as the associated characters. To ) is particularly convincing, and hence we would need . In fact the latter perspective is interesting to explore , d (1) AdS d Γ (2 4.11 was proposed as the dual of the odd-depth truncation of so √ , which in turn could lead to another class of CHS gravity in ) and ( d 4.10 dual of CHS gravity viewed as an exotic partially-massless higher-spin theory in , etc. 1 1 four dimensions, we end up with resulting expression. Using a shift − . It cannot be one of the usual free CFTs as they do not exhibit such operator spectra, − d d d A Zoo of modules In this appendix, wein review terms the of description of the the corresponding various fields of interest for this paper results were announced.Research The Foundation (Korea) research through of theX.B. T.B. grant was 2014R1A6A3A04056670. and supported The E.J. by researchwith was the of the supported Russian Lebedev Science by Physical Foundation theFellowship Institute. Program grant National through 14-42-00047 The the in National work Researchthe association of Foundation Ministry of TB of Korea was (NRF) Science supported funded and by by ICT Korea 2018H1D3A1A02074698. Research Acknowledgments X.B. thanks M. Grigoriev fordescription illuminating discussions of on the shadow field-“Gauge/Gravity versus fields. Duality group-theoretical 2018” (W¨urzburg,30 July X.B. - 03 is August) where also some of grateful the present to the organizers of the conference parity-breaking CHS gravity inthen odd speculate dimensions. whether one can Lastly,to iterate assuming the CHS this construction [ works, gravity one inAdS might AdS with the kinetic operator CHS gravity in AdS.theory The third in question certain is odd whethercan CHS dimensions. be gravity realized can It as behigher-dimensional a is defined Chern-Simons Chern-Simons well as actions theory. may a known prove local Therefore, useful that for it three-dimensional the is local CHS natural realization gravity to of some inquire whether the The truncation to MHSCHS fields algebra would compatible not with be thefields truncated consistent may spectrum, since but lead there the is to truncationCFT no to a subalgebra all consistent odd-depth of theory the AdS [ but their exotic cousins might be decent candidates: in [ clearer guidelines for thebetter regularization understanding from of other inspections. CHSmassless gravity higher-spin This theory from naturally in the calls AdS in point for its of a own right view and of theregravity are can an several afford amusing “all-depth” a questions. partially- truncation The like first the question one is whether from CHS Weyl gravity to Einstein gravity [ None of the results ( JHEP11(2018)167 ) Y + r ) t ( p (A.4) (A.5) (A.6) (A.1) (A.2) (A.3) ) (A.7) , . . . , s by r and sub- 1 s p,t p,t , ) module of Y d Y = ( ( +1 the irreducible , , . . . , s p Y k of gauge param- so s +1 classes of “gauge s p p,t − ⊕ p r =1 , s p,t . Y p k t X s ; Y )] (2) . t r ; − − th ↑ 6 ) p t so ) p r + t d ) + , t 1) reducibilities, which are , s ] for more details). When ( p ) 6 − t ,p ( p − , . . . , s 1 ∆ ) 1 t p,t 111 n p ∆ ( p , . . . , s – +1 V Y p − , Y +1 ; (∆ D 1 the Young diagram describing the ; p ) , s t 106 , t )] t ) ) ( p − and ( r t k ( p + 1 t, s ( p,t p,t − th ↑ ) − ∆ Y p t p,t Y p p − ( p ξ the module defined by the lowest weight (see [ Y := ∆ t p , s V ∆ , . . . , s 1 Y ∇ . This irreducible module is defined as the , s 2 p,t t − 1 ϕ D p +1 ] for more details on the relevant characters = − p , . . . , s − p = Y p , s k,p – 27 – p ϕ current: n 105 ξ − , Y δ , . . . , s d , m d − 1 ; )th 73 , . . . , s s ) is reducible, i.e. it is trivial for some particular class ) k 1 ↑ 1 t + ( p − s ;( , . . . , s = 0 p p − 1 p ∆ ( s A.6 s Y k − ϕ := ( − D ;( := p d t ) p,t t , s 2 p,t ( p + 1 − Y p − ] (see also [ m p s k [ − − − p 2 d 104 , ], by its maximal submodule. The latter is the module whose lowest ∇ + Y p 64 ; , is also a traceless and divergenceless tensor with symmetry s , . . . , s ) [ t 1 ( p 57 s p,t gauge field module / CFT Y ξ ] with = ( +1 ) p,t 1, the gauge symmetry ( d k ( p,t Y th of these reducibility parameters, they can be expressed in terms of ; Y parameters”, the genuine one withobtained diagram from the BGG resolution.k Denoting p > of gauge parameters. This implieseters that is the itself module a quotient,submodule obtained by describing modding the out parameters of a leading generic to partially-massless trivial as gauge we are transformations. considering For here, there are modulo the gauge transformations where ject to a wave equation similar that of can be realized asand the solution space to of the traceless wave and equation divergenceless tensor with symmetry The resulting quotient, The presence of thisare submodule subject signals to thet fact gauge that symmetries, these namely partially the massless submodule fields of lowest weight [∆ corresponds to a partially-masslessand mixed-symmetry minimal field energy with ∆ spin quotient of the generalizedweight Verma [∆ module inducedweight by reads the AdS • discussions in [ from a different perspective). do so systematically, we go through the BGG resolution of these modules, following the JHEP11(2018)167 ), , ) x (A.9) A.11 (A.11) (A.12) (A.10) obeying q, ( ba d boxes are F P t − ) = x ( ab ) 1). Applying the ) d F k ( ( p,t − so Y p χ s k,p 1 (A.8) ν + , which are traceless tensors − k 1 ) p + t ) the character of the module t ( p d − + . The conservation law obeyed 6 ) t p Y ( p k . ∆ 6 , q 6 0 k k,p 1 ν +1 1; (0)) is the image of the trivially conserved ≈ k 6 + − 2; (1)) correspond to the image of the divergence , 1 1) . In this context the submodules involved k d Y − ( − d J +1 p,t ( ( t , + j D 1 ∂ Y D t − 1)th one) until this row has one less box than =0 − p p k,p k X = 0, the minimal energy of these reducibility s + − n . p,t – 28 – ) p Y T t − ,p ( p p,t k 0 P =1 ) + j j ν X Y ∆ − x as follows: p ( ) s = := 1)) corresponds to an antisymmetric tensor d ( , Y so Y := (0) p,t ; k,p χ ) Y ν 3; (1 t ) j,p ( p = 0. Defining t ( p − n ∆ ,p ∆ d 0 ( q th one) in the original Young diagram (i.e. n D D 0. Its first submodule is th row, one then obtain the first reducibility parameter by remov- p p 2; (1)). , i.e. operators of conformal weight ∆ ≈ Y − ) = ab case. The first submodule is spanned by the conservation law ( d x F ( vanishing (i.e. for symmetry reasons). b 1)), while the second submodule D q, is a traceless projector onto +1 ∂ , -dimensional point of view, these modules describe (partially-)conserved ( d ) 21 d in Y p,t ; 3; (1 ) Y T ) ab t ( p P F ,d − b (2 d (∆ ∂ ( so D a D ∂ χ in with the convention that gauge field module in the AdS i.e. a suitable projection ofremaining a submodules (or correspond several) to divergence(s),identically descendants and of all the its conservation descendants. law The whichWe can are now write down (irrespectively of the parity of by these operators is of the form where in the definition of this irreducible module are interpreted slightly differently than one and so on. From the currents of spin with the symmetry properties of the Young diagram Schematically, each reducibility parameter ismoving more obtained and from more the boxes.removed previous from Starting one the from by the re- gaugeing parameter, boxes where from the rowthe above one (the below ( (the same procedure to the obtained reducibility parameter, one can construct the next with again the conventionparameters that reads with by convention with For instance, the module ab F 21 the conservation law of quantity JHEP11(2018)167 Y . ) (A.20) (A.18) (A.19) (A.14) (A.15) (A.16) x (without q, ( c d Y P ! . It is obtained ) . Y x )] (A.13) )] ϕ ( , r r ) without any trace- ) 1 d s k ( ( p,t Y − so Y +1 . d t χ ) ) , . . . , s r which is obtained from + − k,p ( c +2 p ν , p s + Y 1 k , s − , − + , . . . , s r t (A.17) + ) +1 Y +2 + 1 ) t , p p . x φ ( p t s ↑ ( c of the gauge field ∆ ) , s +1)th − ∆ , . . . , s p d q c + ( ( = 0 = 0 − p Y k ) ) t so Y +1 + 1 t p ( ( p p R added by the number of derivative acting , s 1) χ t and projecting them so that the resulting ↑ p ∆ ∆ ) p,t p,t Y − +1)th t ∂ Y ( p t, s Y − ( p Y the module whose lowest weight reads ϕ ( ξ 1 ≡ ξ ∆ ϕ p t t c − : =0 − q 1 Y p ∂ k X + 1)th row , s ∇ p , . . . , s p P +1 − – 29 – − p 1 + d , s Y T this module corresponds to the space of solution ) Y s p 1 = P x P d ( c − − ) 1; ( Y d , . . . , s p d ( c R 1 − so Y s + p χ the finite-dimensional module whose BGG resolution con- c − +1 = ( ∆ p , . . . , s + 1 to the ( d q c s 1 t + 1 derivative of Y
t + = − − − c 2 +1 )-weight, the conformal Killing tensor module has lowest weight s +1 p ) = ∆ ) p +1 s , d d x [ ;( p s s 1 (2 q, s − ( − ) so p is the projector onto the symmetry of the Young diagram c p − s , this corresponds to the space of solution of projects onto the symmetry of the Young diagram s Y ) ; c c [1 ,d Y T Y +1 d P P (2 (∆ so D χ In terms of where lessness condition, whereas inof M the conformal Killing equation Conformal Killing tensor: tains the above gauge field moduleIn corresponds AdS to that of its conformal Killing tensor. of the parity of curvature tensor is given by i.e. the minimal energy of theon gauge it field to make up the curvature. The character of this module reads (irrespectively Schematically, the curvature is given by where any tracelessness condition). Accordingly, the minimal energy of the module of the object as the symmetry propertyby of adding the Young diagram corresponds to the module ofby taking the curvature Curvature of the gauge field in AdS • • JHEP11(2018)167 , # ) ) r s x (A.27) (A.23) (A.24) (A.22) ) ( ) ) x + d ( m ( ) ( c, , projected k + so Y ) − Y . d χ ,..., p ( , φ ( p,t, ) , so Y 1 ) x +2 r m,p )] (A.26) χ , ¯ ν m r s − 1 s − + − p k,p c p +1 q − d ) +1 ∆ ( p − ) , s d ) q ν x ) r r d ( − s + ) ) ) ( − k m,p k − ( , 6 ) +1 x − − − n 1 d d ( p p ) ( j ) ( p,t, − + d + ¯ − r so Y t +1)th ( ( + ↑ 6 χ + ) + ( ,..., (A.25) m so Y , ) +1 1 ) t ,..., x + χ (A.21) ( p + 1 derivatives on ( +2 k,p p m . Schematically, we have x ) ) ( ) +1 ∆ p Y c t t − + − ( p p d , , . . . , s q p m q, φ p ( Y ) = ( c, ν , s d ∆ ( − c t,s ) so Y x q +1 − j,p )) ∆ − p d k χ r − ¯ n p +1 − ) q, s + 1 of the Fradkin-Tseytlin module, which + ) p ( ,s t p t − s ( p t, s 1 ↑ +( c +1 , . . . , s m,p =1 )) m − d +1)th j ∆ ν − r X Y t ) ,p − − p s − p ∂ +¯ 1 ( − C − p p c ) p ( n t the module defined by the lowest weight +1 ) + ( := s ( p c ∆ d , s ,...,s ) , s x : Y T ∆ ,..., − + ¯ 1 p ( = 1 − d d ) P − – 30 – ( − m,p , +1 ) components, this character reads q d c d 2 p ( ¯ 1 ν − +2 − d q M = ,s p ∆ − ( so Y p 1 t,s r +2) c χ +1 , s − 1 d − , − so , . . . , s Y p ( 1 ) p 1 − ) t s C + ,...,s t ( p so ( ,s k s ⊕ ( p 1 + 1 m ∆ χ + 1 +1 ;( 1) , . . . , s − t p − ↑ p 1) 1 d (2) − +1)th = +1 t,s +1 − q ( p − p j ( − so − ( p s = 0. Its character coincides with that of the finite-dimensional ,...,s + p p 1 =1 p 2 1 p k X ,s , s − ,p − s − =0 1 1) p p , s ), i.e. 0 2 " X 1 s − − ν m x − ) − p r ;( , j + 1 1 x − 1 ) s + +( + p 1 − ,d ,...,s q, − p [ s q , . . . , s 1 ( s (2 ( (1 1 d − s so D 2 P s := χ ;( 1 = ( ) representation ) s j,p d ) ,d − ¯ n m (2 (1 ( c, so D Y (2 + χ corresponds to the on-shellis Weyl obtained tensor by acting withso ∆ that the resulting tensor has the symmetry of On-shell Fradkin-Tseytlin field in evaluated in ( and by convention ¯ so with where Written in terms of its • JHEP11(2018)167 . ) c m , ( c Y Y . !# (A.30) (A.31) (A.32) ) ) # x ) x ( ( x ) ) ) ( + d ) m d ) ( k ( c, ) ( d ( p,t , k ( so Y . As already ( p,t so Y ) ) χ so Y )-weights Y χ x dimensions, we x ( d χ ) ) ( d q, + k,p d m,p m ( ( ¯ ν ν k,p ( c, so ) ν + so Y − − c k + ; χ Y k ) ; ∆ + t ) t ( p + q t t ( p + m,p ) ∆ ) + ¯ ν ∆ t − ) ( p t ,d − − ) ( p − c ∆ d (2 , ( x denotes any Young diagram ∆ q ∆ d ( , q so U p q ) ) s ˇ − d Y χ +1 m ( ( c, ˇ k Y +1 − so Y k − ) + 1) t is a field of conformal weight χ ) x η σ 1) − ( x Y + ) ) ( − + : − d m,p 1 ( ϕ p m q, ( ν r ( c, 1 ( =0 − should be understood as a collection of whereas so Y p,t +¯ p k d X =0 − = c p k Y X ˇ χ Y P ) ξ = 2 ∆ Y t σ ) t ( p − d ) + x d m,p ∂ ∆ the off-shell Fradkin-Tseytlin, or shadow, field ) + ( ν q x ) ( x +¯ : d Y ) − ( c ( – 31 – d d ) P m ( ∆ d so Y d ( − so Y 1) χ = d so Y = ) χ , so that − t q σ χ ( p ) Y ( , Y t ) ξ 1 ( p φ ∆ t δ ( p m − ∆ − ∆ and with conformal weight =0 p ∆ d q 1) X − q q m Y
r − ) is meant to represent the fact that the shadow field is a (A.29) + ( 2 " (A.28) 1 Y ) − ; − ) := ) = ) x =0 p t x ( p ) = x X − m q, r x ], the corresponding character can still be computed by translating q, ( ∆ q, ( d " ( ) q, 64 ) ) − P ( + 1: Y c ; ) x ) Y c d r ; t ( p c Y ) ; q, ) is the traceless projector onto the symmetry of the Young diagram ∆ , subject to the gauge transformation c ∆ ( project onto the symmetry of ,d ,d = 2 d − ) − ∆ c Y (2 ( d (2 Y Y T P ,d − ( d so D d P P so S (2 ( χ χ The first term, whichweight ( is the charactertensor of of symmetry the generalized module with lowest so D χ associated to the (partially) massless field – Y where: and consequently is notas found proposed in in the [ BGGthe field-theoretical resolution definition in of even thistherefore dimension. conformal define gauge field. However, In even where obtained by taking aWeyl trace gauge of parameters withmentioned the previously, this symmetry field of does all not possible correspond traces to of a generalized Verma module and spin Notice that in bothcorrespond cases, to characters the of generalized modules Bianchi labelled identities by verifiedOff-shell by the Fradkin-Tseytlin the field in Weyl tensor. M φ and for The character of this module reads, for where • JHEP11(2018)167 ), − ) . p,t, (A.37) (A.35) (A.36) ) x # Y . ( ) x ) ) ; ( # + d x m ) ) t ( ( ( c, + d ) , m ) ( so Y ) ( c, − d ) k ( ) x χ ( p,t so Y . t ( x ( p so Y ) χ ) ) d χ k x q, ∆ ( m,p case, except the ( p,t ( ¯ ν ) is irreducible in m,p ) so Y − q, r k,p − ¯ c ν ). This module ap- ( Y χ ν d c Y ) − ; ( − ; + ∆ c ) Y c ; t k V Y q k,p = 2 ) ∆ ( p ) ∆ t ν ; + ( p q t ) ,d − d + ∆ t ) d ∆ ( p + k (2 ( ) + ) ,d − t , + − ∆ so D ( p ) + x d t (2 ( ) ( ) d χ ∆ + x ) ) − c ( ) so D ( q − Y + 1 case can be describe in a d t d m ) ) ( ( p Y ; − χ ( c, D ( − d r ) ; m ( ∆ so Y ) t ( c, +1 c U ( p − q k χ so Y x (A.33) ∆ ) ∆ χ = 2 ) 1) q, +1 x ( x m,p − k d − − ( ν d ( q, m,p ) d d 1) ( +¯ 1 ν P d ( ( c ( d =0 − +¯ ) − D V ∆ c p k X so Y ( P x 1 − ∆ ) ( χ d ) =0 − ∼ = − ) x d q t p k X d ( ( ( p ) ) + ) q ∆ so Y x d Y ( m ( ; − – 32 – χ ) ) ) + d m so Y t ) d 1) t ( p ( q x χ ( p 1) ( − " so Y ) ) can be computed following the algorithm spelled ) ∆ ∆ ( t ) − d ( p χ 1 x − ( ( x ) − d ∆ − 1 t so Y =0 p q, q ( p − d − q, ( ). χ X d =0 − ( m p ∆ ) ( , the shadow field module in odd dimension is isomorphic r ) q q X case, by defining d − t m − D ( p (A.34) r 2 ) = Y P − + r ; ∆ ) " x A.31 t q ( p ) ) := ) = q, ) ∆ = 2 x x x ( ) = ,d − ) − d x q, d Y q, q, ), we can now write explicitely the character of the shadow field (2 ; ( ( ( ( ) ) d ) t q, so U ( p ( Y Y χ P ; ; ) ) ∆ ) ) t t − A.33 ,d − ( p ( p Y d ) ) ; (2 ∆ ( ∆ ) t ,d ,d − − so D ( p ], which yields d d (2 (2 ) χ ( ( ∆ 64 so S so S ,d − + 1. As a consequence, this module is now part of the BGG resolution in the d χ χ (2 ( r The second term is the characterpears of in the the module BGG resolution asand the corresponds maximal submodule to of thegauge pure symmetry gauge ( modes of the shadow field associated to the so U as χ – = 2 Y where one can recognize thedenominator module of corresponding the to above the equation. off-shell Weyl Its tensor character in therefore the reads The two terms heremodule have of the the same pure interpretationd gauge as modes in of the thesense shadow that field it is isomorphic to the quotient identical. It is worthsimilar noticing way though to that the the As pointed out in section to that of the off-shell Weyl tensor in odd dimensions, and hence their character are φ Upon using ( The character out in [ JHEP11(2018)167 for . ` 2 ) (A.41) (A.42) (A.38) (A.39) (A.40) − x 2 , +1 1 d − q ( ) − Y ; . ) ) are part of this ) t ` ) used throughout ) ( p ,d x (2 (∆ 1.2 )). q, so D ( 6= 0; χ d 2.7 r , and Di P s − ) ` ) x ) x x q, ( ( q, ) ) ( c d ) only if ( − Y + ; − Y so c r Y ; Y χ ) ) ∆ + ) t = as in the bosonic case). ) integral dominant weight and ( p ,d Another class of irreducible modules, − ,d ∆ Y for bosonic irreps, or ∆ = d d . ` s , (2 k ( − ( (2 ∆ ; (∆ d 0 so D so D + q so ∆ ; − χ ), one can see that the three character, for χ ≈ ` d − 2 V with Y ) , . . . , r 2 obeying to a wave equation of the form: ) = φ (A.43) ) + D − x ) A.29 ( x x ) we recover the expected result d ) – 33 – Y x 2∆ ∼ = = 1 d q, q, ( − + ( q, ( d ` , . . . , r ) Y so ) ( ∂ c Y Y A.35 even one can consider more general conformal fields χ )) ; ∆ = Y ) r ; ) and ( t s d c ∆ = 1 ( p ) is given by singletons (respectively, Rac of spin q − ∆ ; ) ) an arbitrary ) ∆ with ∆ k r ` ,d ,d − − Y s ` ,..., d A.21 d D 2 (2 (2 ( φ ( 2 A.39 for − +1 so S so D +1 ) = p d ), ( d χ k χ ) x s t,s Now that we have written down the character of the modules − q, , these fields do not enjoy any gauge symmetry (contrarily to, for ( − , these two families of singletons exhaust the fields described by p ( +1 0 , A.12 ) d ,s p k 1 Y 1 ) (i.e. the spin of these fields). It is clear from the above conditions 6= − 1) − ,d r p 1 2 (2 − Y (∆ ; ], as their conformal weight read so D + 1: ∆ = = ,...,s 31 : ∆ = χ r 1 = ( r s − , . . . , s 2 1 s s = 2 = 2 ;( 1 ) d s d = ( ,d − = 0 or (2 (1 , are related by For For fermionic irreps (with the same condition on in accordance with the factof that the the off-shell module Weyl of tensor. the shadow field is isomorphic to that and hence using the definition ( so D r s Y χ • • = 2 the paper. Combining ( d Notice that even if instance, the usual CHS fields sourcing theCharacter conserved relations. current in ( involved in the description of CHS fields, we can derive the relation ( this class of module.(in However, particular for they can bebe of realized spin as greater conformal or fields equal to one). In general, the above class can with that in odd dimension The character of the module ( The scalar and theclass spinor of order- module [ with Series of modules withwith non-integral non-integral weight, weight. can exists and are defined by the quotient JHEP11(2018)167 . . ) ) x ) . , with 2) = x (B.1) c 1 , , . (A.48) (A.44) (A.45) (A.46) Y w − 1 A.34 (2 ], where q V C − ( ) d 2) Verma q ) so 21 , ( − x − ) c , Y (2 Y 1 ; ; ) ) ) 2∆ ) − t t and conformal so ) and ( ( p ( p ,d ∂ q ,d ( c (2 (∆ ) (2 Y (∆ Y , Y so D so D ; ) ) A.29 ) χ t χ ( p x ,d x C d ), ( (2 (∆ )+ q, d ) + ( so D x ) d x χ Y q, 1) Verma module ; M ) A.12 ( ) q, , t Z ( ( p +1 )) ,d ) d c (2 r , and that the integrand has c (2 s (∆ ∆ Y Let us conclude this appendix 1) Y so ; − so D − c ) − χ t ) ∆ ( p ,..., . Accordingly, the spin label can s ,d = 1 conformal algebras: − ∆ − 2 d − +1 (2 ( ∆ ) + p d x (A.47) 1) ) + ( so D x ) d t,s − x χ x − q, + ⊗ V p ( q, x q, ) s ( ,s ), also leads to ( c ) = ( 1 2 +1 Y Y )) p ; ∆+ − ; r ) c ) ) p s t V ( p ) is derived by requiring it to be conformally ) ∆ ,d Y − A.43 = 2 d – 34 – +1 ,d − φ (2 d (∆ ,...,s d d (2 1 ( so 2 ) = D 1) − − s so D s A.48 ) S − ) = ( 2 t ( χ ( p s x ;( 1 ,..., 2∆ (∆; as d ) and are related to the q, ) s +1 ∂ ( ) = s p V +1 ,d − 2 p )) x (2 1) (1 r Y TT t,s so D q, (∆; − − P ( χ ) V p ,...,s Y on M Y ,s ; +1 = ( 1 +1 ) t p p ( p − the character of the Weyl module is identical to that of the shadow x φ p ) x t,s ∆ 1) d d ,d ] for the case of totally symmetric fields). − − d − p d (2 ,...,s d ( 95 ,s 1 . A careful analysis of such action principles can be found in [ so S 1 M − d χ − Z 2 as p s ) = ( 1) corresponding to left and right movers if we write the flat metric in . Schematically, it takes the form ;( , x ) w 1 t ) s ( p ,...,s (2 q, 1 ,d − ] = ( ∆ + 1 however, the relation reads ) − Y so (2 (1 2 Y r φ ; − s so D [ ) ⊕ ;( t χ d ( p 1 = 2 conformal algebra is a direct sum of two ) s ) = 2 ∆ 1) ,d d CHS − ,d − , d d S (2 (1 (2 ( (2 so D so S χ light-cone coordinates have positive/negative real values correspondingmodules to left/right will chirality. be The denotedlowest as weight invariant (see also [ B Two-dimensional case The so Notice that the number of derivatives involvedthe in this latter action can to be constrained beconformal by weight requiring quadratic in thein shadow particular field the with form spin of the action ( Action principles inby the commenting metric-like on the formulation. weight action principle for a free conformal field of spin The above relation allowscharacter of to the express associated theconformal Killing partially character tensor massless of (depending field, the on plus the shadow symmetry or field of minus in these the fields). terms character of of the their Recalling that for odd field, we can write the following relation valid in any dimension: For which, upon using the previously derived ( χ On top of that, one can derive the following relation between ( JHEP11(2018)167 ) s := are − = 2. ; 0 ) (B.8) (B.9) (B.7) (B.2) (B.3) (B.5) (B.6) ) s . The d w ( s ( = 0 (so w ϕ V − ⊕D (∆; J 1) describe ) + V s ]) so we will ∂ ) one finds the ; + 79 + 1) and we made 1, in accordance (1; s , ( + D ), where (2 J . D 2.30 s > − ) x so ¯ ∂ z ( := s ) s − 2 0 , − ( ) ∆ ϕ , . s (B.4) ¯ z i α ; ) ) s 1 )(1 . , 1) Verma module − s s z z = 2 and = z , ( ) ) β (¯ ( 2 ⊗ z z d − ∆+ D 1) 1) (2 ⊗ V (¯ ( − s , , e z V 1 1) 1) (2 (2 so , , s s (1 = ··· V so V so (2 (2 s s J ∼ = ∼ = 1 χ χ so V so V − ) − − χ χ ) = s 1) z q x (¯ = = = = ) ) − − s 1) s s , = − 2 s 1 1 ¯ ) is given by z z ) in each sector separately, then one gets − (2 ¯ ∆ z s s s s s ; + ; − − so V ¯ − − ¯ z z z z s s ( ( χ ¯ z z 1 1 + 1; + 1; 1 2.16 (∆; ) V V – 35 – s s = 2. By formally applying ( z V ) which both solves the Klein-Gordon equation , ( ( )) are ill-defined for ( d V V ) = ) = s ) = ) = 1) i α , x 2 s, s ( + (2 ∆+ ) for the character of the shadow field by the analytic β q, x q, x q, x q, x so V (1) ( ( ( ( − describes the conformal spinor while ) for ) := ) := = 0. This leads to ) ) χ ) ) (2; ( vanish, in which case the conservation condition becomes e ϕ s s s s s s 0 2.39 D 2) 2) − ) − , , − ;+ ; ;+ ; = 1 2 = s s s s ; ; + 2.30 −··· 2) 2) ) = (2 (2 ( ( , , ; s s − − −···− + ( ( so D so D 1 2 (2 (2 q x (2 (2 J J ( χ χ J q, x D D 1 . Accordingly, their respective characters are so S so S − The modules of conserved currents remain well-defined for ( = D χ χ ) ∂ traceless conserved currents. Indeed, the tracelessness implies that ∼ = s 2) z , s 1 in one dimension described as the (2 (∆; V ]). ). so V / w s 0 The modules χ − V 112 = 0 or + (∆; 2) weights, the character of 2, one can identify the parity-invariant modules ··· , 2 require a more careful discussion (see e.g. section 4.2 of [ ) and + > ⊕ V (2 J = 0 (so it is a submodule of Rac) and the conservation law ) s − B.1 so s ∂ s < J For − ∂ (∆; + + continuation of the identityintriguing ( relations it is a submodule of the completeShadow fields. module describing the spin-1with conserved the current). fact thatone Weyl tensors might are replace identically the vanishing formula in ( two dimensions. In such case, densities of weight spins just mention that Dia = left/right chiral∂ boson The above can bee.g. straightforwardly section generalized 6 to of [ the partially-conserved currents: (see as describing the spin- all components of the type either use of ( for the two chiralities, where 1 stands for the trivial representation of The chirality-invariant modulesV will be denoted withConserved 0 currents. as subscript,In particular, e.g. if one applies the definition ( for the Introducing the complex variables, JHEP11(2018)167 ] 1) , 113 = 1. ) ) = 0. 1 s (2 x ) − ) − ) s ) (B.11) (B.10) s 1 2 1 0 s 1 so h V V ⊗ D ⊗ ⊗ V ⊗ ⊗ V ⊗ V 1) highest- , (1 (1 (1 (1 (1 (1 = 2. (2 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ . Its character d ] by setting so 1) 1) 1) 1) 1) 1) ∆ 64 V ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ s s 1 0 1 1 2 2, one can first reach − V − V V V s 1 ( ( ( ( is the V D > Equivalent descriptions ( ( ∆ s V 2 in a topologically trivial vanish. The residual gauge = 2 case (cf. the appendix of [ > d s −··· . field current of spin ) + . s s q s h ) ( 2 h s 1) 2) ∓ , , ; 2)-modules and their field-theoretical s (2 (2 ( , s CFT 1. so V so D (2 Conformal χ 1) is non-local. Nevertheless, its partition Shadows of χ ⊗ chiral bosons , Killing tensor Chiral bosons so − − ∆, which we will denote = 2 Flato-Fronsdal theorems (cf. section 4.2 = 2 and the negative sign corresponds to the Conformal spinor 1) Shadow spin- ) . This is nothing but the higher-spin version − − = d d = 0 s ∓ ( ) = ) Conserved spin- – 36 – s . s x 1 − ( − ) 2 ; 1. In this sense, one can write the formal equality: V s q s 2) Conformal higher-spin gravity is highly degenerate ( , − − − and allows to fix the reach the trivial gauge ]). As far as characters are concerned, one can write 1) (2 (1 ∓ , (2 s s ) = 2) modules and their field-theoretical interpretations so S − ϕ field field s , (2 1 χ s s so V 113 − (2 = , ··· 1) lowest-weight Verma module ; χ 3 ξ , s so Consider a shadow field 64 ( (2 Di S ∂ ) . Hence, it may be tempting to reinterpret the character ··· AdS 22 1 Chern-Simons Chern-Simons so ξ = − 2 2 shadow fields in terms of the one for the contragredient q ⊗ ⊗ and Killing tensor ( ). They rely on the = 1, one can obtain the relation s 1) , 1) d Massless spin- Massless spin- ··· U(1) U(1) − (2 with Dirichlet behavior with Dirichlet behavior ∆ s h 2.41 with Neumann behavior with Neumann behavior ( so V ξ δ − χ 0 ) for V . List of relevant 1) ⊗ 0 0 0 0 0 − ) ) 2.24 ) ) = 1 2 s s s q ; ; ; ; ( s s 1 2 s ( (1; 1) ( (1; 1) ( 1) ) = 1 , − Table 2 S S D D s D Modules (2 ∆ (1 The list of the relevant parity-invariant Using ( so V ; + This formula is in perfect agreement with the standard treatment of D χ s = 2 versions of ( 22 ( background while the sectorfunction of low can spin be ( definedd (see [ for a detailed discussion) and in particular reproduces the partition function given in [ S interpretations are summarized in table Conformal higher-spin gravity. in two dimensions since its action identically vanishes for The contragredient of the weight Verma module withis highest weight formulae for the spin- Verma module with highest weight of the usual infinite-dimensional enhancement of conformal symmetries in residual gauge symmetries. the traceless gauge wheresymmetry all reads components ofEven the after type this complete gauge-fixing,functions the of parameters one of variable: the residual gauge symmetry are which one can summarize as In fact, the shadow fields are pure gauge in JHEP11(2018)167 ) ) s 2.18 (C.1) B.10 (B.12) (B.13) (B.14) ; (C.2) ) 1 − r )] r . Using ( )-module into a s ,...,s , d V 1 s , . . . , σ / on-shell FT module . 1 (2 s σ 0 s − . so 1 [∆ ; ( 1) 0 ;( V ) − V 2) n , s,s s,s ∼ = d (2 ( . 2) , − R )-module of the on-shell FT so D 0 1 ∆+ ) (2 (1 A 2) χ , − acts trivially) of highest weight , d so D s )) s,s V (2 ( d d χ (2 =1 ∞ ( D r so D s X ) which we omitted for simplicity. R s so so 1 χ =1 ∞ = ⊕ s − + X 1 r − M 0 s =1 1 2 ∞ (2) r B.13 s X 1) σ so − = ( + . By summing over all spins, one can say 0 ··· s,s ) 0 U 2) 0) 2) 2) 2 ) , , , , − 2) s , ⊗ s,s 1 (2 (1 (2 (2 (1 ( s,s = s (2 ( 1 according to the following branching rule so D M so D so D ) σ so D – 37 – χ χ χ χ , d 1) =0 ∞ =1 =1 ∞ n ∞ M − (2 − module (on which s s X X = 2 0 ) so ) 2 − = , d d ( s,s 1) 0 ) U 2) (2 = ) 2) , − , − ,d 0) on each side of ( so 0 (2 , ↓ so ,d (2 (2 s,s , i.e. we decompose this irreducible ]. ) (2 := 2) ⊕ , so S r ]: − (2 so Di (1 so s,s χ (2 2) χ (2 so ) 45 , generalized Verma module − D , (2) r so S (2 = 2 ) (2 115 χ = 2. so , so S 40 d 1)-modules. The latter can naturally be interpreted as fields in , d , d = 2 χ ) =1 ∞ r s 0 X (2 − 114 39 , . . . , s =1 ∞ 1 0) s so X is an s , , d ) can also be derived from the fact that the finite-dimensional spin- )] into a product of kinetic operators of partially-massless and massive fields (1 (2 branches onto r , . . . , s d r 1 The D so )] ∆ ; ( s B.13 r ,...,s 1 = 2 s V d ), one deduces ∆ ; ( ) also holds in obtained in [ [∆ ; ( , . . . , s d V V 1 1.1 For ]). For instance, in the simpler type-B case, = 2, i.e. the difference B.13 , thereby allowing us to recover the factorization property of the kinetic operator of s 1)-module can be seen as the following quotient: d , d • 79 (2 [∆ ; ( the following lemma [ Lemma C.1. where CHS fields in M in AdS C.1 Branching ruleIn for order generalized to Verma derive modules the branching rules for on-shell Fradkin-Tseytlin modules, we first need In this appendix, wefield derive in the even branching dimension ruledirect for sum the of AdS as the analytic continuationfor of the onethat for ( the quotient in the right-hand-side ofC ( Branching rule for the Fradkin-Tseytlin module so and ( One may define formally the character of the parity-symmetric spin- Strictly speaking, the relationinvolve for a the term full spectrumNotice of that type-B off-shell ( CHS theory would This implies the relation in [ JHEP11(2018)167 , d ] ). Y R [∆; (C.6) (C.7) (C.3) (C.4) (C.5) V ) while , . . . , d . r in which ) defined , ) d r Y d = 1 R R , . . . , s A (2) acts diago- 1 (∆; to a direct sum ) as ∼ = a, b s V which is an irre- so Y d )) } , . . . , σ p ] a d 1 ]) is simple enough, ( Y = ( σ P the translation gen- (∆; of { so ;( Y ] a V [∆ ; 115 Y P ⊕ n , V , [∆; , we only need to consider i 1 (2) 114 V = span ∆ + − Y so d . Due to the fact that the p 1 +1 . V ] g − , r ∆ ; d Y s . In order to obtain the branching | := ( ] ) is induced from representation of p − in the sense that r +1 n Y and s [∆; d = a ) is then constructed by acting with g , d M d p r d V [∆; p , Y σ p (2 ⊕ } V on the finite-dimensional module ⊗ ) is then defined by the action of the 0 so ...P ··· ) g Y (∆; 1 ) 2 ,D a s V +1 ⊕ 1 ab , d P = g s 1 1 ( M (∆ ; (2 M σ − the dilation generator (while = U V { g – 38 – annihilated by special conformal transformations so n =0 ∞ ∼ = D n M Y ...a U ) = 1 The generalized Verma module a Y , d i = span 1) ) (2 Y − 0 ,d ∆ ; ; ) acts through the representation g ↓ ,d so (2 d n ( (2 V so (2) is present in both so so so , ∆ + d | carries a representation of . In practice, the action of (arbitrary powers of) the generators ) R defines new vectors (or states) in the generalized Verma module, together with the Lie bracket between generators of this subalgebra d ] r ] p ] Y ) generators and ∼ = ) acts trivially. In other words, it defines a primary field, i.e. a field Y d Y a ( [∆; } [∆; a [∆; K V so . In more concrete terms, we can write a basis of is trivial. Concretely, the conformal algebra admits a three-grading de- , . . . , s V + 1 V K d 1 { d p r s stands for a generic basis element of ), the first task is to branch the representation on R the on i span( Y d = 2 d p are the special conformal transformation generators, ab Y ∆ ; ( . In other words, the representation of ∼ = d R (∆; a M = span 1 +1 V ) is constructed as follows: one starts from the module ∆ ; V ∼ = − 1 K g | g For − g C.1 The idea of the proof (for more technical details, see e.g. [ +1 • ∼ = g d where rule of of representation of itspart lower acts dimensional trivially counterpart, and the The action ofsubalgebra the conformal algebraand on of the subalgebra the whole universal enveloping algebra modulo the action of of so that The vector space nally with ∆ asR eigenvalue, with fixed conformal weight ∆ andat spin the origin. The generalized Verma module where erators, the action of composition so that (as a vector space), with before giving another proof in terms ofPoincar´e-Birkhoff-Witt characters. basis. by ( ducible representation of the homothety subalgebra when expressed in terms of the basis, Poincar´e-Birkhoff-Witt so that we recall it here, JHEP11(2018)167 i )– P (C.8) (C.9) C.8 (C.11) (C.12) (C.13) (C.14) (C.10) . Then the in the above σ . , σ ) , ) 1 ) (2)-eigenvalue ∆ x − r r so q, ( , , ) )) ) on the direct sum of ) r 1)-weight Y d x ( , . . . , σ + 1), the branching rules ( R , . . . , σ 1 − ) has weight +1 under the ( stands for a generic basis ,...,σ 1 1)-weights r ), and commutes with any σ r 1 d σ ( U d i ∈ B σ ( 1) (2 ( r − +1 , 1) s ;( r P − g σ s ) ∼ = ,...,σ σ n so d so d − − r 1 1 ( ( = s x ) , ,d = σ − 1 M so ( r r ∆; i so − (2 M , s q, (∆+ σ | +1 χ r 1) generalized Verma modules. To ( and labeled by the V so 2 g σ σ 1 N r ( − ) and ; χ 1 − s ] n − r r , so that the remaining generators 1 − ∈ U n Y d s − r r r d s − s 2 , d = s = P (2 r X ) now reads 1 P r [∆; − − M r s = r q (2 σ s − 2 so = Y M s V X r ∆ + r 1, while − σ ) can also be recovered at the level of char- 1 so − | r σ ··· σ 1 − (∆; 2 , say ∼ = ··· s C.2 ··· 1 d V 2 i , n, p s ··· = 2 +1 – 39 – s X s σ 2 1 1 i g ) = σ )–( = s s s = 1 1 X M x = -module σ σ s σ 1 1 M ∆; (2)-weight ∆ + ) = σ q, C.3 | − ( . Due to the fact that ∆; =0 x d ∞ so d n | ( X n p Y ) ) ) P ) designates the set of n r d r +1) 1) ) ) Y r P ↓ r d (2 − ( ) = ( ) r (2 ↓ with P ,...,s (2 d so ). B x ( ( 1 (2 so so . The basis of 1 s run from 1 to p so ( i 1 so q, − ) irrep into a direct sum of ) of the conformal algebra result from the branching rules ( ( p C.3 χ d − d d ) component. For both p ) )) ( 1 r r d R ) , and − C.2 ...P ( so r d σ 1 p i so ,...,s , . . . , i )–( 1 P 1 s ) i 1), we have , . . . , s ,d + 1, we have 1 C.3 , . . . , s (2 − s (∆;( The branching rules ( 1 1) span r ( so V s d ( ( − χ 1)-weight = 2 so , d d − d ··· ( , so that we readily obtain in accordance with ( and For so ) of the rotation subalgebra. • = 1 i C.9 Characters. acters as follows: i.e. it defines anbranching irrep rules of ( ( element of the finite-dimensional and branching rule of the adjoint action of the dilationelement generator of (as any generator of ( where the indices The next thing tothe do representation is of to decomposedo the so, action let of us single out one generator of are well known and read respectively and the branching of the JHEP11(2018)167 , ) r , d ( ) 1 so − r (C.18) (C.19) (C.20) (C.21) (C.22) (C.15) (C.16) (C.17) 1) is x , + − ) k r x , d ( , ) (2 1 ) r − , . . . x so , . r x ) 1) 1 +1 r 1) ) q, − − k , ,...,σ 1 ( x d x 1 1 ( ( − r ) character to 1, say )) σ ) − 1 so ( x 1 d r + ( , − − χ r r ) + r so +1 1) ˇ s x r k , 1 ( − = ,...,σ , ) ,...,σ ) − d 1 1 r 1 r ( , x r ) , . . . , x s σ X − s σ r 2 1 r so ( 1) ;( − , σ x , x n χ − 1 1 , . . . , x of the 1 ,d − − 1 q, r r s ··· − k ( − 1 (2 1 , . . . , x x := ( (∆+ 1 2 x = x − k s so V 1 ) − 1 r ˇ x ,...,s d s d X − = s χ + ( 1 + X r , x σ s P 1 σ r 1 so ( 1 1 s ) − q − + 1 whereas the rank of x χ − = k k x ( r ··· r r X ( 1 s − δ σ ) 2 r s 1 1 ) = ( k s = with ˇ X ··· x ) is A σ Vandermonde determinant. In order to sim- – 40 – ( 2 , . . . , x , . . . , x ) s , d ) = 1 1 r 1 r r s =1 = ˇ − 1 ,s x x X ) = k (2 X σ 1 x , × ˇ x − q, so r ( ( = ( r + ) =0 ∞ d k,r ) = r X n 1 k δ s ) P x ,...,s d x x ( ( 1 , ( ) s ) is less straightforward to recover at the level of character 1 ) simplifies to so ( δ r ) = ). − s χ ˇ r ) x ) = 2 − r C.2 s ˇ x q, ) ) C.2 ) is the C.15 ( ( ,..., − k ,...,s d d ) r ( 1 ( 1 x )) r s s ( k r so ( so ( + χ χ A r ,...,s s k 1 x s ) ) + ,...,X ) which at the level of characters reads 1 ,d ), one can set one of the variables x , the rule ( ( r (2 X (∆;( ) ( r so V ) = ( C.15 δ χ C.15 x = 2 ) ( ,...,s d ) ( 1 d r s ( k so ( , so that the characters on both sides of this equations depend on the same number r A χ in accordance with ( and hence Similarly, we have and the branching rule ( where plify ( x of variables. Due to the fact that and where due to the factand that hence the the rank charactersthe of characters modules of of the the former.irreps latter This ( depend can be on already one observed less for variables the than branching of For • This concludes the alternative proof via the computation of characters. JHEP11(2018)167 s . ) ) x ) x (C.24) (C.25) x q, q, ( 1 dimen- ( q, d 1 ( − dimensions P 1 − ) as defined . 0 d d − d ) d ) (C.23) Y P r ; P x 0 ); x ( ) ) ) x x x q, (∆ m − ( ( ( 1 q, ) ) 1 dimensions (given V ( )) d 1) m ( s d ) 1), we can deduce the − − s,s, 1 t ;( , so ( d P s d partially-massless fields 1) ( − − χ s − +1 − so ( t 1) ) t t in ) to each of the terms ap- m ,d χ , d − x − t d − ( (2 2 ( (1+ FT field in odd 2 (2 s,s ) − 1) so S so ( − d d FT field in even ( so − C.14 d χ χ ) are those of the modules de- s + q r so ( s ), the characters of such quotient ), and reinterpreting the resulting q χ x m ) + 1 Y r ) + − − q, A and depth- − 3 ( x off-shell x ) d ) and ( − ( s )) and depth- x + ) d (∆; s q, on-shell s ( q s ( m + q D 2;( 1) 1 ) )) − s d C.22 − − − ( t d 1) s,s, ( ) ) 2;( m − so ( s − d − so ( x χ t ,d 2+ ( + 1) χ ) s q − – 41 – m (2 ( d − 2 d ( ) ) are the characters of the corresponding spin- . Notice that the former/latter modules can also so D ,d − s + r 1 2+ m t so ( s χ (2 q ( x − − χ 1) d d so D 2 q, + 1)-module of the − χ m ( − s ( d 4 )-module of the )) q 3 − 1) 2 s − + =0 As emphasized in the rest of the paper, the module of cen- − d s ;( , d − =1 ) defined in terms of quotients of generalized Verma mod- r X s t m + , d q ( X 1) + at hand, we can then obtain the branching rule of an irre- Y (2 fields, namely as spin- s − 2 − t (2 2 =0 − ,d d so r X m − so ) = as explained in the previous paragraph to the character of the (2 ). (∆; (1+ C.1 ) = x so S yields, after taking care of a few cancellations Y D ˇ x χ q, r ) = C.1 ( , the characters q, x ( (∆; )) = 2 q, D shadow fields in M +1)) ( t ) d s,s t )) − ,d 1) , . . . , s in the rest of the section): ) (2 − s,s (2;( )-module s,s )), while ,d r ,d so D , d = 1 (2 (2;( χ (2 (2;( so D (2 t D so = 2 3.18 χ χ d so ). Before deriving its branching, let us spell out its character in all dimensions ), while the branching rule of such a module translates straightforwardly at the level scribing partially-conserved currents of spin- in ( and depth- be interpreted as AdS with Dirichlet/Neumann boundary behavior. For The character of the sions reads The character of the reads 2.18 • • • C.1 The characters appearing onfollowing the modules: right-hand-side of the above equation correspond to the Now applying Lemma Weyl module for tral interest in CHS gravityin is ( that of the(taking on-shell (totally-symmetric) FT field introduced of its character. Hence,pearing applying in the the character identity of ( expression an as irreducible a module sum ofbranching characters rule of for irreducible modules of Fradkin-Tseytlin modules. Having the above Lemma ducible ules. Indeed, as weare have given seen by previously an (in alternatingin appendix ( sum of characters of generalized modules C.2 Branching rule for irreducible lowest-weight modules JHEP11(2018)167 . d + ]) d 1). (3)- 38 − + 1) so (C.27) (C.28) (C.26) t , while , d 2 d − (2 . so ) FT field, the s s, s with minimal ,..., t , ;( 3 as a module of d 2; ( n d ) − σ − D d 3 ;( ) s τ = − ] (conjectured in [ . d − + ) 2; ( ) are those of the modules 45 with two possible choices )) and s , r D s − . Firstly, the left-hand-side d x ;( + 2 5 t 40 t =0 − q, , s 2; ( , σ d n M − ( − 39 − )) d into a product of wave operators ⊕ D s is the conformally-flat sphere S ··· t t d , + 2;( + + 2 R − s s +1 − s t massive fields on AdS d d − σ 1) = 1 − σ M s − d t + , we can write the branching rule: D = ,d + τ s ⊕ D AdS More generally, for a depth- s s,s (2 ( ( ], let us rephrase the heuristic argument as s =1 t +1 C so D M S 23 t 65 χ s − . Therefore, defining a conformal field on S as modules of the isometry algebra s 1 M + 1) d = − – 42 – we need to sum over the two boundary condi- 4) 3) t with minimal energy ∆ , , σ d ↓ d (2 (2 d − are absent since finite-dimensional irreducible so so 3) 4) , , s, s -dimensional conformally flat space M ↓ (2 (2 1)-modules d ) so so +1) 2; ( s − t 1. These modules come by pair of conjugate dimensions ∆ − D , d , the characters ). Secondly, the right-hand-side correspond to its description s, 4 − = 4 is special, as no massive nor shadow fields appears in the (2 2 s − , d =1 s, s t 2 d d is conformal to one hemisphere of S M d corresponding to massive field with identical mass but Dirichlet so (2 and correspond to the off-shell FT module either in terms of the 2 ; ( d + 1)) + 2; ( ) are the characters of spin- + so 1) t ) r D − ∆ ,d x ,..., = 4 reads D − ↓ ,d (2 − d s q, (2 so ( ( , . . . , s , . . . , s = 2 d so )) or in terms of its Weyl tensor s − +1 = s ;( = 1 s, s s 1) ) h t − − − t = ,d 2 ; ( s, s t for the partially massless fields of depths (2 3)). (1+ , so S D d 2; ( describing massive fields onχ AdS energy ∆ and ∆ vs Neumann boundary behavior. For (2 Notice that the case D so We preferred the Euclidean signature because the geometric picture is more intuitive, but it holds in • 23 branching rule for Lorentzian signature as well. In fact, the modules modules labelled by two-row Young diagramsfor vanish (and hence these modules do not exist Higher-depth Fradkin-Tseytlin modules. tion choices. branching rule, so that ary choice. Following similar observationsfollows: in [ the conformal boundaryIn of turn, Euclidean Euclidean AdS of boundary conditions atin the terms equator of S the one on Euclidean AdS in AdS describes the on-shell FTthe field conformal on algebra in terms of partially massless fieldsThirdly, each on partially AdS massless field gives rise to two modules: one for each possible bound- This decomposition is inwhere accordance the decomposition with of the the results CHS wave of operator [ in Having in mind that the coincide for shadow field JHEP11(2018)167 (C.29) ) σ ) σ 2; ( . Using Weyl − 2; ( = 4 maximal- s τ d µ − p − τ ]. Several qualita- d ··· − 29 1 + d , µ ]). The kernel of each p σ + 28 ) is conveniently packed ) 41 x σ ] for the concrete example x ) ( ( s σ s ⊕ D , corresponds to the module h µ 116 ;( 4 ··· n ⊕ D 2 − 1 ) modules whereas the kinetic d µ 2 − ) 6) the branching rule also involves h 2 − . For instance, the + 1) 3 σ + d s > t τ ;( ) − P + r σ σ − − d t s − 2; ( ) = = 2 D − τ factors (see e.g. [ matrices. This formalism is very useful for σ, σ 5 d − s 2 =0 − n ( k x, p d n 2 M − t ( 2; ( 1 + × d − h t – 43 – n +1 and the CHS action is the Seeley-DeWitt coeffi- D S t + t t − s . The weak field expansion of the latter coefficient − t (see section of 3.4 of [ 4 + + s d M s s ˆ 2 − s ··· H = − d − − + σ σ σ σ M M s + s + + = = σ ⊕ 2 s τ τ The tower of shadow fields p µ ˆ p ranging from 0 to ⊕D +1 +1 = ˆ t t k s s − − ˆ ) ··· G s s M M 1 σ = 1 case), namely = = µ t σ σ ;( p s ) ˆ ]. In higher dimensions ( x − ( 66 k s µ + 1) + ··· t t 1) ) 1 − µ ,d − h ↓ ,d ∼ = (2 1+ s (2 ]. As such, it remains a somewhat formal definition which becomes concrete so ) FT fields have two-derivative kinetic operators but their spectrum is made of s, s t so S P ] of the operator 28 ˆ = , G 2; ( = 2 case). [ +1 s t 2 d t 27 ) = − D a ˆ p s = = x, s s σ (ˆ ˆ brevity) which consists in writingbols all and formulae in treating terms these of operatorselucidating operators as the rather large structure than of their the sym- fragile. theory Only but the its perturbative field-theoreticalin formulation interpretation with momenta is symbols admits sometimes admitting a power clear series interpretation expansion as a local field theory. From the point of view H cient can be computed via standardtive features techniques of in nonlinear quantum CHS mechanicsthe gravity [ are “formal more operator easily understood approach” within (which what will Segal called be shortened here to “formal approach” for only once it is computed perturbativelyflat in vacuum the weak solution. field expansion around the conformally- Formal operator approach. in a generating function overcalculus, phase the space: latter can be interpreted as the symbol of a Hermitian differential operator D A primer onThe nonlinear nonlinear CHS action gravity ofpart type-A of CHS the gravity effectivefields is [ action defined of as a the conformal logarithmically scalar divergent field in the background of all shadow factor operator,L labelled by depth ( maximal-depth PM fields of spinof 1 to Notice that thisoperator branching for rule the contains FT 2 field contains that this is in accordanceFT with fields the derived factorization in of [ themore partition modules function (as of in maximal depth the i.e. partially massless fields of different spins (as well as different depth) appear. Notice JHEP11(2018)167 φ ) of ∞ (D.4) (D.5) (D.6) (D.1) (D.2) (D.3) ( u )-singlet N ) of infinite C , ∞ ( which are gauge gl ˆ G ) which is quadratic of infinite sequences limit of the U( as the kinetic operator 2.7 ∞ n ) on its Lie algebra φ C C -filtration of vector spaces: . In other words, “formal , N 2 ) form a group of invertible ∞ 1 ... D.3 GL( 0 in a suitable dimensional regu- − . ˆ ˆ ⊂ → , i O O ) → φ | +1 = = ∞ d ˆ ). Therefore, the formal approach is N , † † G C | i ˆ ˆ . 24 ˆ O O O, φ φ ˆ | h → X ˆ , i G 1 ) are actually generated by two Hermitian , and to higher-spin diffeomorphisms + † N − ˆ ˆ S S ˆ ] = C ˆ O O e e D.3 N ˆ ˆ O, O, ∈ → – 44 – is the direct limit of the = = s → stands for its Hermitian conjugate. The spectrum ˆ ˆ } . G G ∞ ˆ ˆ † s i → ..., O O 1 ˆ ˆ C ∞ ˆ h G O φ O ]). − | { C → , ˆ ; 92 O Using Weyl calculus, the action ( 2 → φ On the one hand, it is not clear whether such invertible [ C ˆ → G S → as follows: , 26 ˆ . G C ˆ ˆ X X i ) and they couple to the free scalar field e carries the fundamental representation of the infinite general linear group )-vector model should be replaced with the large- = ∞ and D.3 C N ˆ ˆ O S -vector (for instance an element of the space n ). )-vector model (see e.g. [ ) form a group because the products of such operators may not be the expo- -dimensional CHS theory (and, similarly, of its related Neumann MHS theory equivalent to ignoring functional and locality issues, as well as polynomial +1 d d nN D.4 25 is an invertible operator and ) which one can describe as the group of infinite invertible matrices with finitely-many non- ). In the formal approach, the gauge symmetries ( ˆ O C , D.1 Notice that the space In mathematical terms, the vector space More precisely, the U( ∞ 26 24 25 sector of a U( GL( vanishing complex entriesmatrices outside with the finitely-many non-vanishing diagonal. complexmay entries. correspond Its to In particular, Lie theinfinite the adjoint algebra Hermitian higher-spin action matrices is diffeomorphisms of with the the finitely-many non-vanishing real Lie entries. subgroup algebra U( The collection ofcountably-infinite the basis sequences of the with vector one space entry equal to 1, and all other entries vanishing, provides a for invertible and Hermitian operators for unitary operators operators ( They correspond, respectively, to higher-spin Weyl transformations where of off-shell CHS gravity isfields with spanned symmetries by ( Hermitian differentialin operators ( operators. These finite gaugedifferential operators symmetries ( It is manifestly invariant under the gauge transformations larization of around AdS Formal group of symmetries. in the conformal scalar field can be rewritten in a compact way as as if it were awith large finitely-many non-vanishing complexessentially entries UV-divergencies, in the field-theoreticalCHS discussions gravity” of should section correspond to the limiting case of the effective action, the formal approach amounts to treat the conformal scalar field JHEP11(2018)167 )– ) is ) and (D.7) (D.8) (D.9) D.2 (D.10) (D.11) x ( D.8 2 − s ) of shadow σ 2.2 between shadow fields As a corollary, the s j s 27 x h ). In fact, the spectrum d and (ii) given modulo the d ) with tensors ˆ p 2 R D.9 p x, (ˆ = ˆ such that ˆ , 0 X 0 − ˆ − ˆ G ] ] O ˆ ˆ X X , , . i ) and 2 ˆ 0 order in the derivatives). This subtlety 0 ˆ φ p G, p | [ [ˆ ˆ ˆ O G i ) i x, ˆ (ˆ ˆ H = X + + ˆ i S † 0 0 finite + + ˆ ˆ ] ] O + O ˆ ˆ S S 0 – 45 – ˆ S , → ˆ ( 2 G ˆ G, p † − 0 ˆ H ˆ By definition, the global symmetries of CHS gravity O The infinitesimal form of the transformations ( = = [ = [ˆ i ˆ ˆ G H φ | are linear transformations of the perturbation δ δ δ 0 modulo the gauge equivalence ( ˆ O denotes the (anti)commutator. The linearization of ( operators (i.e. of ˆ H ˆ ) reproduces the form of the gauge equivalence ( A CHS gravity is also a module of the higher-spin algebra (because . The tower of currents spans the twisted-adjoint module. Each shadow field ˆ B s D.9 ). j ) be the Weyl symbols of ˆ D.8 B ˆ x, p A on-shell ( CHS gravity is a module of the higher-spin algebra. differential ξ = off-shell CHS gravity can be defined as the space of Hermitian differential off-shell CHS gravity (i.e. the tower of all shadow fields) can be encoded in a which are (i) in the vicinity of the vacuum ] ˆ ˆ G B ) and ˆ A, off-shell ) as coefficients in the power series expansion in momenta. Then, the infinitesimal x, p ( x ) is ( σ Another simple way to reach this conclusion is to consider the pairing 1 nonlinear and conserved currents linearized 27 − s s D.3 the tower of Bachthus tensors the quotient span is a also submodule a module). isomorphic to theh twisted-adjoint module, is the dual moduledual of of the the current twisted-adjoint module. with respect to this pairing, therefore the tower of shadow fields is the since they preserve theof vacuum (C)HS by algebra definition. onthat Its the the infinitesimal gauge spectrum version equivalence of definesof relation the free free is action off-shell preserved CHS byspectrum gravity. this of free action. In fact, Therefore, one the spectrum can check The generators ofgenerated such by operators such operators span the (C)HS algebra. The finite transformations gauge symmetries ( Algebra of global symmetries. preserve the vacuum, i.e. they correspond to operators fields (up to signs andof factors which we omitsingle for simplicity). Hermitian In operator otherof words, the spectrum operators Let ξ gauge transformation ( where [ somewhat precludes a mathematically rigoroussymmetries. definition of On the the other group hand, ofbecause the higher-spin infinitesimal differential gauge transformations operators are form perfectly an well-defined associative algebraAlgebra under of composition gauge product. symmetries. ( nential of some JHEP11(2018)167 ) ). i λ ( D.3 F except (D.17) (D.14) (D.16) (D.12) (D.13) i x P )] = ˆ G ( through its sign. . F , while the invari- λ )] ˆ F G z in this region, then the − . 0 (except maybe for the , )] i > ˆ G ˆ ] (D.15) ) G z | → ∞ z ˆ t G z Tr[exp( − [ ) can rescale the eigenvalues by | n ), i.e. the adjoint action of uni- | − z . a dz i D.5 , i.e. n )] D.6 1 2 t ih ˆ I G i ) was argued by Segal to imply that exp ( | even, one can say that the coefficient of Tr[exp( =0 = ∞ 2 1 h πi 1 X n i ) can only depend on c d 2 z D.5 2 d )] for some function λ dz λ Tr ( ˆ − G + t t i ( F dt 0 + ] = X 0 F ˆ G 2 ∞ [ 1 – 46 – ∞− = Λ 2 d )] = ] := Tr[Θ i ∞− = 1 and Z i are real numbers. Therefore, Tr[ ˆ ˆ a ˆ − U G G [ + i t − C is the left-continous Heaviside step function (i.e. it ˆ Z ] =Tr[ G λ − ] = ˆ 0 1 G Segal CHS [ ˆ πi − 1 G [ S ) is the discontinuous function vanishing for all 2 ] := ˆ U x ˆ CHS G sgn (where “sgn” stands for the sign function). [ Heat log 1 2 W ] = Tr[exp( A standard field-theoretical regularization of the one-loop ef- ] which must be invariant under the gauge symmetries ( ˆ W )] is analytic in the region Re( G + Heat Λ [ ˆ ˆ G G 2 1 2 d [ related to the value that one assigns at the origin to the Heaviside ) where Θ ) can be modified as W z a x c ] := − CHS ˆ G ∆( [ ∆ = W D.16 c 1 2 In the formal approach, the action of CHS gravity is obtained by look- CHS , the heat kernel expansion takes the form: to be determined. Transformations ( and the eigenvalues + S = 0 where it is equal to the unity. The heuristic argument is as follows: ) + ˆ 1 0 H F x − x ( + ˆ 0 U 2 Θ := Θ p = are the Seeley-DeWitt coefficients. For and a constant C 1 2 † is Hermitian, it can be diagonalized by a unitary transformation, = ˆ n ˆ a U C ˆ ˆ G G ) = x ( of heat kernel Tr[exp( neighborhood of the origin) andcontour integral approaches to in zero ( when the logarithmically divergent term identifies with the nonlinear CHS action: which can be seen as contour integral around the origin of the heat kernel. If the trace if one ignores IRform divergencies. For a semidefinite Hermitian differential operator of the where Heat kernel expansion. fective action is the heat kernel prescription: function. Segal considered the choice where Θ where for a function positive factors, hence anThis invariant essentially function leaves the Heaviside stepfactor function as only possibility up to a normalization vanishes at the origin) whileat ∆( the origin since ing for a functional The invariance under higher-spintary operators, diffeomorphisms ( implies that ance under higher-spinF Weyl transformations ( Formal action. JHEP11(2018)167 under in the (D.20) (D.21) (D.22) (D.18) (D.19) ˆ ). How- G ˆ G ), which is z of eigenval- taken as an ˆ 1) . Indeed, D.13 − 1 ˆ n G x ˆ 1) — under the ) for individual − Tr( 2 0 x D.19 → z = ] with Tr( and Tr( ˆ q G [ ) 2 d . , ˆ ) and ( G ) a ). One may still think that ) 0 stating that the Hermitian ˆ , but we can reorganize them G zλ Θ( 2 δ D.5 . ] , the logarithmically divergent ≈ x − D.18 ˆ . ˆ 0 G G ) [ ) z i — is because the operator 2 d ] in the zeta function regularization − λ exp( and zλ a ( ˆ e i ≈ . G ) δ [ 2 1 − 0 ˆ 1), namely the number 2 d ˆ z x G x dz | Tr( a z ˆ + G i ≈ 0 dz + exp( − z 2 0 )] with i − | I e ˆ z ∞− G dz ) | i ∞− ( exp( 1 ˆ ˆ 1) is by regarding it as lim i G ) would lead to the operator equation 1 2 − – 47 – πi 1 x ( I + 2 h δ Z . But in fact both of them are very formal expres- | − ) and the relative position 1 1 π i dz 2 D.13 i ) 2 1 π i h x ˆ G 2 I ) = Let us consider the equation of motion of CHS grav- + ), in agreement with Segal’s formal action ( ˆ G Θ( 1 πi 1 = 1 ( x ) = 2 ( λ 2 1 D.17 ( CHS 1 2 = δS Θ x ), then we would be lead to consider the identity function instead D.16 in the formal action ( ˆ G ) as ), one finds the collection of equations is non-singular in the sense that it has no normalizable eigenvector with van- ˆ G D.16 D.12 ˆ 1) looks more weird than Tr Using ( operator ishing eigenvalue, as was argued by Segal. This is indeed the case for the vacuum solution The above equation isinto bilocal the depending center on position both auxiliary variable generating infinitely many higher-spinthe fields. operator On the other hand, varying we obtain the equation of motion as consideration is positive semidefinite whileof the zero contribution measure. of the zero modesFormal is probably equation ofity motion. derived from theaction action ( principle. On the one hand, varying the operator regularization is invisible ineigenvalues. the Another way integral to representation regularizeknown Tr( ( to give the logarithmically divergentmethod. term The reason foraforementioned the conditions formal imposed equivalence between on Tr Tr Then, we would identifyeven the in logarithmically the divergent formalues, term is viewpoint invariant taken under byTr( higher-spin Segal, Weyl transformations Tr( ( sions as they give divergent series when summing over the eigenvalues. The corresponding ever, one needs to bewith cautious here. the trace For instance, in ifof ( we the boldly Heaviside: permute the contour integral Therefore, it is tempting topiece of identify Tr[Θ the effective action ( The Heaviside step function admits an integral representation of analogous form, JHEP11(2018)167 ] ] ]. Int. , ]. (D.23) Int. J. (1989) , SPIRE ]. IN 50 [ SPIRE IN ) by a (large) [ ˆ O SPIRE (1985) 233 IN = can be mapped to arXiv:1702.08218 hep-th/0406192 [ is a lower-triangular [ [ † 119 ˆ G ˆ ]. ˆ L L JETP Lett. , (1919) 101 and SPIRE (1989) 97 59 hep-th/0605089 0 , IN (2017) 538 (2005) 147 ˆ G ][ Parent field theory and unfolding ]. Phys. Rept. = , 260 B 231 ˆ 1). Their equivalence indeed relies ˆ D B 921 SPIRE ]. IN , † Annalen Phys. ˆ L , ˆ D SPIRE Phys. Lett. ˆ L , hep-th/0504090 IN [ Nucl. Phys. [ – 48 – ≈ , (1988) 33 [ ]. ˆ Formal higher-spin theories and Hochschild cohomology of the Weyl algebra and G 36 Cubic interaction in conformal theory of integer higher Conformal supergravity SPIRE Commun. Math. Phys. (2006) 37 , IN [ ), which permits any use, distribution and reproduction in 3 in the close vicinity of the vacuum solution keep this prop- ˆ H + In the formal approach, one may then use the Cholevsky decom- 0 arXiv:1705.02958 ˆ Fortsch. Phys. , G , (1991) 1115 = CC-BY 4.0 Higher spin algebras and quantization on the sphere and hyperboloid Actions, charges and off-shell fields in the unfolded dynamics approach Extended higher spin superalgebras and their realizations in terms of Quantization on sphere and high spin superalgebras ˆ Off-shell gauge fields from BRST quantization G This article is distributed under the terms of the Creative Commons ]. A 6 A new extension of relativity theory ]. ]. ]. SPIRE IN can be any chosen diagonal matrix with positive entries and in Euclidean signature since there are no normalizable solutions of Laplace equa- SPIRE SPIRE SPIRE 2 ˆ IN IN IN D p Vasiliev’s equations [ spin fields in four-dimensional space-time Kontsevich–Shoikhet–Tsygan formality [ in BRST first-quantized terms [ J. Geom. Meth. Mod. Phys. quantum operators 374 [ Mod. Phys. E.S. Fradkin and A.A. Tseytlin, E.S. Fradkin and V.Ya. Linetsky, M. Grigoriev, A.A. Sharapov and E.D. Skvortsov, A.A. Sharapov and E.D. Skvortsov, G. Barnich, M. Grigoriev, A. Semikhatov and I. Tipunin, M.A. Vasiliev, M.A. Vasiliev, M.A. Vasiliev, M.A. Vasiliev, H. Weyl, = ˆ [7] [8] [9] [5] [6] [2] [3] [4] [1] 0 [10] [11] ˆ any medium, provided the original author(s) and source are credited. References the vacuum solution ingauge the transformation. formal approach (setting Open Access. Attribution License ( position of positive-definite Hermitian matrices where invertible matrix with positive diagonal entries. Therefore, on-shell tion. Therefore, one may alsosince argue all that operators the formal equationerty. of This motion agrees is somewhat with empty theon alternative ignoring formal the action contribution of Tr( zero modes. Formal gauge fixing. G JHEP11(2018)167 ] B ] ]. Phys. ]. , JHEP B 829 , SPIRE , (2002) 1376 ]. SPIRE IN Phys. Lett. IN , Annals Phys. Mod. Phys. Lett. ][ 133 (2016) 118 , , hep-th/0207212 SPIRE [ ]. 12 IN Nucl. Phys. ][ , arXiv:1701.08645 [ (1990) 293] [ SPIRE -dimensions JHEP IN ]. , (2003) 59 (2013) 667 198 ][ hep-th/0103208 (2 + 1) [ SPIRE (2017) 054 ]. B 664 IN Theor. Math. Phys. B 876 , ][ 04 5 ]. S arXiv:1506.03328 SPIRE [ × Annals Phys. IN 5 [ [ JHEP (2001) 1155 SPIRE ]. , ]. arXiv:1207.3439 IN Nucl. Phys. AdS Nucl. Phys. [ [ , The Schouten tensor as a connection in the , 128 ]. – 49 – SPIRE Effective action in a higher-spin background (2016) 142 Higher spin conformal symmetry for matter fields in SPIRE IN (1989) 731 (1991) 274 IN [ arXiv:1312.5883 Conformal superalgebras of higher spins 08 ]. Superconformal higher spin theory in the cubic Conformal superalgebras of higher spins A superconformal theory of massless higher spin fields in SPIRE [ ][ Cubic interaction in extended theories of massless higher On the gravitational interaction of massless higher spin Conformal higher spin in IN (1987) 141 A 4 Notes on the ambient approach to boundary values of AdS Higher order singletons, partially massless fields and their ][ ]. ]. ]. (2013) 214008 B 350 SPIRE ]. JHEP Conformal gravity in three-dimensions as a gauge theory , IN (1987) 89 ]. ]. [ B 291 (2015) 078 SPIRE SPIRE SPIRE A 46 SPIRE IN IN IN Theor. Math. Phys. [ 09 IN , ][ ][ Bosonic Fradkin-Tseytlin equations unfolded. Irreducible case Bosonic Fradkin-Tseytlin equations unfolded SPIRE SPIRE B 189 ][ On the conformal higher spin unfolded equation for a three-dimensional Towards an exact frame formulation of conformal higher spins in three IN IN conformal higher-spin fields Nucl. Phys. arXiv:1012.2103 [ [ On limits of superstring in Bosonic conformal higher-spin fields of any symmetry , [ (1989) 501 D Mod. Phys. Lett. J. Phys. 3 JHEP Conformal higher spin theory arXiv:0909.5226 , Nucl. Phys. , [ 62 , ]. ]. -dimensions Phys. Lett. (1989) 2363. , (1989) 245 (1990) 252 (2011) 048 = (2 + 1), SPIRE SPIRE arXiv:1305.0162 hep-th/0201112 IN IN arXiv:1412.7743 gauge fields boundary values in the[ ambient approach [ [ 02 unfolding of [ arXiv:1807.00142 dimensions [ self-interacting scalar field (2 + 1) (2010) 176 D Rev. Lett. 225 spin fields fields approximation A 04 198 X. Bekaert and M. Grigoriev, X. Bekaert and M. Grigoriev, A.A. Tseytlin, A.Y. Segal, X. Bekaert, E. Joung and J. Mourad, T. Basile, R. Bonezzi and N. Boulanger, O.V. Shaynkman, O.V. Shaynkman, B.E.W. Nilsson, O.V. Shaynkman and M.A. Vasiliev, M.A. Vasiliev, B.E.W. Nilsson, J.H. Horne and E. Witten, C.N. Pope and P.K. Townsend, E.S. Fradkin and M.A. Vasiliev, E.S. Fradkin and M.A. Vasiliev, E.S. Fradkin and V.Ya. Linetsky, E.S. Fradkin and V.Ya. Linetsky, E.S. Fradkin and V. Ya. Linetsky, E.S. Fradkin and V.Ya. Linetsky, [30] [31] [27] [28] [29] [25] [26] [23] [24] [20] [21] [22] [18] [19] [15] [16] [17] [13] [14] [12] JHEP11(2018)167 , , ]. , , , SPIRE ]. J. Phys. (2016) ]. structure IN , (2014) 066 (2013) 016 (2018) 080 ][ ∞ 06 10 L (in Russian), 01 conformal SPIRE (2014) 734 D 94 SPIRE IN [ D IN ]. 4 (2016) 084 ]. ][ JHEP JHEP JHEP , , B 885 , 12 SPIRE SPIRE IN Phys. Rev. ]. , IN ][ ]. JHEP arXiv:1710.03779 , ]. [ SPIRE Nucl. Phys. IN SPIRE arXiv:1105.5632 , , ][ IN SPIRE arXiv:1707.02456 ]. ][ ]. IN [ ]. ]. (1984) 1018] [ ][ (2017) 082 On the Weyl anomaly of 39 (1984) 307. SPIRE SPIRE 11 SPIRE SPIRE arXiv:1702.00222 IN On triviality of S-matrix in conformal higher IN IN [ IN ][ ][ ][ Scalar scattering via conformal higher spin (2017) 123 ][ – 50 – ]. B 134 ]. JHEP ]. 09 , for conformal higher spin fields from partition function arXiv:1607.06379 T On the structure of the conformal higher-spin wave [ On conformal higher spins in curved background SPIRE Instanton zero modes and beta functions in supergravities. arXiv:1512.08896 SPIRE C On induced action for conformal higher spins in curved (2017) 359 arXiv:1309.0785 Gauge invariance versus masslessness in de Sitter space SPIRE [ IN IN [ [ JHEP ]. ]. A note on higher-derivative actions for free higher-spin fields IN [ ]. ]. ]. , From conformal to Einstein gravity On conformal higher spin wave operators ][ Phys. Lett. Sov. J. Nucl. Phys. , B 919 SPIRE SPIRE (2016) 034 arXiv:1709.00850 SPIRE SPIRE SPIRE arXiv:1209.4864 arXiv:1802.02968 IN IN arXiv:1609.09381 [ [ [ (2016) 125 IN IN IN [ ][ ][ (2013) 598 09 (1984) 396 ][ ][ ][ 02 Worldline quantization of field theory, effective actions and One-loop effective actions and higher spins One-loop effective actions and higher spins. Part II AdS description of induced higher-spin gauge theory Arbitrary spin conformal fields in (A)dS Effective action in de Sitter space and conformal supergravity On partition function and Weyl anomaly of conformal higher spin fields Einstein gravity from conformal gravity 154 (1984) 1606 [ JHEP Nucl. Phys. B 877 Induced action for conformal higher spins from worldline path integrals (2017) 64 , , arXiv:1808.04320 (2012) 153 (2018) 095 JHEP 39 3 , , arXiv:1608.07826 11 04 [ (2017) 125401 arXiv:1306.5242 arXiv:1404.7452 arXiv:1404.3712 arXiv:1609.02088 arXiv:1709.01738 higher spins: a holographic approach exchange spin theory background on conically deformed sphere JHEP [ A 50 Yad. Fiz. Annals Phys. [ operators 2. Conformal supergravity Nucl. Phys. [ JHEP 086008 [ [ Universe E. Joung, S. Nakach and A.A. Tseytlin, M. Beccaria, S. Nakach and A.A. Tseytlin, M. Beccaria and A.A. Tseytlin, M. Beccaria and A.A. Tseytlin, S. Acevedo, R. Aros, F. Bugini and D.E. Diaz, S. Giombi et al., M. Grigoriev and A.A. Tseytlin, A.A. Tseytlin, S. Deser and R.I. Nepomechie, E. Joung and K. Mkrtchyan, M. Grigoriev and A. Hancharuk, E.S. Fradkin and A.A. Tseytlin, A.A. Tseytlin, R.R. Metsaev, T. Nutma and M. Taronna, L. Bonora et al., J. Maldacena, G. Anastasiou and R. Olea, L. Bonora et al., R. Bonezzi, L. Bonora et al., [51] [52] [48] [49] [50] [46] [47] [43] [44] [45] [41] [42] [38] [39] [40] [35] [36] [37] [33] [34] [32] JHEP11(2018)167 ] , , J. 01 , ]. D 74 JHEP (2015) , Class. JHEP SPIRE ]. , , ]. IN A 48 ][ ]. ]. Phys. Rev. hep-th/0401086 , ]. , [ SPIRE SPIRE ]. IN IN ]. ]. SPIRE SPIRE ][ ][ J. Phys. SPIRE IN IN , [ IN SPIRE ][ SPIRE ][ SPIRE IN (2006) 823 ]. IN IN ][ ]. ][ ][ 18 arXiv:0808.3945 ]. ]. [ SPIRE IN Unfolded form of conformal equations SPIRE SPIRE ][ arXiv:1112.5620 arXiv:0907.4678 IN [ IN [ hep-th/0201019 hep-th/9906217 ][ , ][ hep-th/0508031 ]. [ , conformal fields and AdS/CFT (2009) 128 [ 5 ]. Partition function of free conformal higher spin Partition functions, the Bekenstein bound and arXiv:1512.06443 Conformal higher spin scattering amplitudes ]. arXiv:1611.06200 Scattering of conformal higher spin fields arXiv:0709.4392 [ Rev. Math. Phys. AdS [ arXiv:1602.00948 SPIRE ]. [ ]. , – 51 – [ B 671 (2012) 161 IN SPIRE ]. [ IN (1999) 295 SPIRE 06 (2010) 106002 ][ IN Boundary values of mixed-symmetry massless fields in SPIRE arXiv:1307.5043 Iterating free-field AdS/CFT: higher spin partition function On higher spin partition functions ][ IN modules (2006) 062303 [ Conformal higher spin theory and twistor space actions SPIRE (2016) 769 (2017) 021 ][ arXiv:1406.3542 (2012) 062 IN B 563 D 81 [ JHEP 47 arXiv:1604.08209 ]. ][ Phys. Lett. + 2) , 04 [ 06 Conformal and Einstein gravity from twistor actions Boundary action of free AdS higher-spin gauge fields and the , (2003) 5021 (2016) 295401 M B 913 Higher spin gauge theory and holography: the three-point functions ( arXiv:0912.3462 o SPIRE [ JHEP IN JHEP A 18 A 49 [ , , (2014) 113 (2014) 045014 Phys. Rev. Nucl. Phys. , hep-th/0606186 arXiv:1805.00394 Gauge invariant two-point vertices of shadow fields, AdS/CFT and Conformal totally symmetric arbitrary spin fermionic fields Mixed-symmetry fields in Ordinary-derivative formulation of conformal totally symmetric arbitrary CFT adapted gauge invariant formulation of arbitrary spin fields in AdS and Light cone form of field dynamics in Anti-de Sitter space-time and AdS/CFT , 08 Notes on higher spin symmetries 31 [ [ Character formulae and partition functions in higher dimensional conformal (2017) 485401 J. Math. Phys. Point particle-symmetric tensors interaction and generalized gauge principle arXiv:1410.7314 Nucl. Phys. , [ (2010) 115 , J. Phys. ]. , arXiv:1503.08143 JHEP 09 [ A 50 , (2018) 016 SPIRE IN temperature inversion in Anti-de Sitter(2006) space 084009 and its conformal boundary field theory modified de Donder gauge JHEP holographic correspondence 275401 Int. J. Mod. Phys. AdS space theory relations arXiv:1211.4498 (2015) 077 correspondence spin bosonic fields conformal fields Quant. Grav. in M dimensions and [ Phys. from twistor space 07 G.W. Gibbons, M.J. Perry and C.N. Pope, F.A. Dolan, R.R. Metsaev, S. Giombi and X. Yin, E. Joung and J. Mourad, M. Beccaria and A.A. Tseytlin, A.Y. Segal, A. Mikhailov, M. Beccaria, X. Bekaert and A.A. Tseytlin, M. Beccaria and A.A. Tseytlin, R.R. Metsaev, R.R. Metsaev, A. Chekmenev and M. Grigoriev, R.R. Metsaev, R.R. Metsaev, T. Adamo and L. Mason, O.V. Shaynkman, I.Yu. Tipunin and M.A. Vasiliev, R.R. Metsaev, T. Adamo, P. H¨ahneland T. McLoughlin, T. Adamo, S. Nakach and A.A. Tseytlin, P. H¨ahneland T. McLoughlin, [72] [73] [69] [70] [71] [66] [67] [68] [64] [65] [61] [62] [63] [59] [60] [56] [57] [58] [54] [55] [53] JHEP11(2018)167 , 25 , 4S2 ] , JHEP (2002) , , 07 (2018) ]. Nucl. , Phys. Lett. 07 ]. (2017) 066 , B 644 JHEP Fortsch. Phys. (2013) 214013 SPIRE , , 02 IN [ JHEP ]. Conference Moshe , A 46 Lett. Nuovo Cim. Class. Quant. Grav. arXiv:1805.05646 JHEP , , [ , vector model SPIRE Nucl. Phys. ) IN , (1987) 207 N [ J. Phys. ( adjoint model holography arXiv:1402.5396 , O [ ) 109 N (2018) 091 SU( 10 , talk given at the ]. n ]. (1978) 421 ]. 2 The shadow operator formalism for ]. JHEP Partition functions and Casimir energies in SPIRE (2014) 024048 SPIRE , ]. ]. IN IN SPIRE ]. ]. ][ ][ IN SPIRE – 52 – ][ D 90 Character integral representation of zeta function in Character integral representation of Zeta function in ]. action on three-dimensional CFTs and holography scalar conformal field theory IN Comments on higher-spin holography SPIRE ) SPIRE Twisted flato-fronsdal theorem for higher-spin algebras k ][ SPIRE Z IN AdS dual of the critical IN , ]. IN Commun. Math. Phys. ][ ][ Singletons on AdS SPIRE , One-loop test of free ][ Conformally invariant differential operators on Minkowski ]. Free Lett. Math. Phys. IN AdS/CFT correspondence and symmetry breaking , ]. ][ Setting the boundary free in AdS/CFT SPIRE Phys. Rev. One massless particle equals two Dirac singletons: elementary ]. , IN Massless higher spins and holography A note on multitrace deformations and AdS/CFT SPIRE d hep-th/9905104 ][ IN [ SPIRE hep-th/0309177 arXiv:1603.05387 arXiv:1802.03232 ][ [ [ [ IN ]. SPIRE ]. hep-th/0210114 /CFT ][ IN [ [ +1 d arXiv:0805.1902 arXiv:1202.0543 hep-th/0404124 Holography, unfolding and higher-spin theory Higher spin superalgebras in any dimension and their representations [ [ [ SPIRE (1999) 89 hep-th/0203168 IN Multitrace operators, boundary conditions and AdS/CFT correspondence [ [ (2003) 020 (2016) 061 (2018) 009 ]. (2002) 213 : I. derivation of the general formula . Part II. Application to partially-massless higher-spin gravities 12 07 04 B 556 , September 5–8, Dijon, France (1999). +1 +1 arXiv:1805.10092 hep-th/0205131 d d [ [ (2012) 882 (2004) 046 SPIRE arXiv:1607.07439 arXiv:1203.5554 IN (2008) 195014 60 [ JHEP [ Phys. hep-th/0112258 (2002) 027 132 303 B 550 AdS [ AdS JHEP higher spin AdS JHEP Flato 12 space and their curved analogues conformal algebra. vacuum expectation values(1972) and 115 operator products particles in a curved space. 6. X. Bekaert, E. Joung and J. Mourad, C. Brust and K. Hinterbichler, R.G. Leigh and A.C. Petkou, SL(2 M.A. Vasiliev, G. Compere and D. Marolf, I.R. Klebanov and E. Witten, E. Witten, A. Sever and A. Shomer, E. Sezgin and P. Sundell, I.R. Klebanov and A.M. Polyakov, T. Basile, E. Joung, S. Lal and W. Li, T. Basile, E. Joung, S. Lal and W. Li, S. Giombi, I.R. Klebanov and A.A. Tseytlin, J.-B. Bae, E. Joung and S. Lal, E. Angelopoulos and M. Laoues, M.A. Vasiliev, T. Basile, X. Bekaert and E. Joung, S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, M. Flato and C. Fronsdal, M.G. Eastwood and J.W. Rice, [92] [93] [89] [90] [91] [86] [87] [88] [84] [85] [82] [83] [80] [81] [77] [78] [79] [75] [76] [74] JHEP11(2018)167 , ]. , 43 , 4 SPIRE J. Phys. ] 49 ]. B 877 IN boundary , ][ j (2018) 138 (2017) 086 Universe 05 SPIRE 02 , J. Alg. IN , (2016) 096 ][ Nucl. Phys. 05 JHEP JHEP Ukrainian Math. J. , , , , ]. arXiv:0812.4438 duality for spin- (2017) 056 [ (2018) 055 JHEP 4 arXiv:0812.3615 ]. , SPIRE [ 04 04 IN ][ /CFT SPIRE 5 JHEP IN JHEP (2009) 014 arXiv:1508.07332 The analytic structure of conformal , . [ ][ ]. ]. (2009) 013 07 ]. ]. Rainbow vacua of colored higher-spin ]. . 07 SPIRE SPIRE SPIRE SPIRE JHEP IN IN Unfolding mixed-symmetry fields in AdS and the Unfolding mixed-symmetry fields in AdS and the , IN IN (2016) 003 The ABC of higher-spin AdS/CFT Flato-Fronsdal theorem for higher-order singletons ][ ][ ]. [ ]. JHEP ][ – 53 – , arXiv:1511.05975 01 [ hep-th/9810231 Type-B formal higher spin gravity SPIRE SPIRE [ Generalized Killing tensors and symmetry of IN IN A note on vectorial AdS Partially massless higher-spin theory JHEP (1995) 78 math-ph/0506002 ]. ]. ]. ][ ][ arXiv:1611.00112 , Partially-massless higher-spin algebras and their , [ The conformal characters (2016) 150 SPIRE SPIRE SPIRE B 354 05 (1999) 331 arXiv:0910.3334 arXiv:1410.7668 IN IN IN arXiv:0904.2919 [ [ [ ]. ]. ][ ][ ][ 524 (2016) 077 Gauge fields in (A)dS(d) and connections of its symmetryGauge algebra fields in (A)dS(d) within the unfolded approach: algebraic aspects JHEP Weyl anomaly of conformal higher spins on six-sphere Massless mixed symmetry bosonic free fields in d-dimensional Anti-de Sitter Arbitrary spin massless bosonic fields in d-dimensional Anti-de Sitter space Long, partial-short and special conformal fields , 12 Generalized killing tensors of arbitrary rank and order A generalization of the Bernstein-Gelfand-Gelfand resolution arXiv:1310.1795 Phys. Lett. arXiv:1608.07611 [ (2010) 106 (2014) 131 , [ ]. gravity JHEP 01 11 (2009) 385401 , 3 SPIRE IN arXiv:1712.05415 arXiv:1804.03196 arXiv:1702.03938 arXiv:1604.02091 arXiv:1610.08510 JHEP (A)dS (2013) 632 BMV conjecture. II. Oscillator[ realization A 42 Lect. Notes Phys. BMV conjecture: I. General formalism (1977) 496. [ space-time theory finite-dimensional truncations JHEP [ (2018) 18 (1991) 734. Klein-Gordon-Fock equations blocks and the generalized Wilson-Fisher[ fixed points [ [ S. Gwak, E. Joung, K. Mkrtchyan and S.-J. Rey, A.A. Tseytlin, N. Boulanger, C. Iazeolla and P. Sundell, E.D. Skvortsov, E.D. Skvortsov, R.R. Metsaev, N. Boulanger, C. Iazeolla and P. Sundell, J. Lepowsky, A. Bourget and J. Troost, R.R. Metsaev, J.B. Bae, E. Joung and S. Lal, E. Joung and K. Mkrtchyan, M. Grigoriev and E.D. Skvortsov, S. Giombi, I.R. Klebanov and Z.M. Tan, T. Basile, X. Bekaert and N. Boulanger, A.G. Nikitin, A.G. Nikitin and O.I. Prylypko, R.R. Metsaev, C. Brust and K. Hinterbichler, F. Gliozzi, A.L. Guerrieri, A.C. Petkou and C. Wen, [99] [97] [98] [95] [96] [94] [112] [113] [109] [110] [111] [107] [108] [104] [105] [106] [102] [103] [100] [101] JHEP11(2018)167 ]. J. D , Phys. Rev. , arXiv:1305.6040 (2015) 1796 [ Branching laws for verma modules 285 ]. ]. Notes on conformal invariance of gauge fields Adv. Math. SPIRE , – 54 – IN On dimensional degression in AdS(d) ][ arXiv:1506.00595 [ arXiv:0810.2065 [ (2015) 505402 A 48 (2009) 045007 79 and applications in parabolic geometry. I Phys. T. Kobayashi, B. Ørsted, P. Somberg and V. Souˇcek, G. Barnich, X. Bekaert and M. Grigoriev, A.Yu. Artsukevich and M.A. Vasiliev, [115] [116] [114]