JHEP04(2020)206 Springer relevant for April 2, 2020 April 30, 2020 : 2 March 25, 2020 : : with ascending February 10, 2020 2 : isometry subalgebra 2 Accepted Revised Published Received ] and parameterized by a real pa- holography λ [ 1 hs Published for SISSA by https://doi.org/10.1007/JHEP04(2020)206 duals of the kinematical structures iden- . On the one hand, the spectrum of local 1 λ 2  1 CFT c / 2 higher-spin gravity is a theory of an infinite collection of 2 . On the other hand, the higher-spin fields arising through [email protected] λ , and Xavier Bekaert . 3 a,b ] algebra do not propagate local degrees of freedom. Our analysis of λ [ 1911.13212 The Authors. hs ] with conformal weight ∆ = c Higher Spin Gravity, Higher Spin Symmetry λ

[ We aim at formulating a higher-spin gravity theory around AdS , hs [email protected] . The singleton is defined to be a Verma module of the AdS λ ⊂ 1) , (2 Institut Denis Poisson, Unit´eMixte deUniversit´ede Tours, Recherche Universit´ed’Orl´eans,CNRS, 7013, Parc de Grandmont, 37200 Tours, France E-mail: I.E. Tamm Department of TheoreticalLeninsky Physics, ave. P.N. 53, Lebedev Physical 119991 Institute, Department Moscow, of Russia General andInstitutskiy Applied Physics, per. Moscow 7, Institute Dolgoprudnyi, of 141700 Physics Moscow and region, Russia Technology, b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: the gauging of the spectrum suggests that AdS massive scalars with fine-tuned masses,fields. interacting Finally, with we infinitely discuss manytified the topological in holographic gauge the CFT bulk. rameter so modes is determined bysingletons. the It Flato-Fronsdal is theorem givenmasses by for expressed an the in infinite terms tensor tower of of product massive of scalar two fields in such AdS Abstract: holography. As a first step, wecousins investigate of its the kinematics standard by higher-dimensional identifying structuresgleton, the in low-dimensional the higher-spin higher-spin gravity such symmetry as algebra, theparticular, the sin- the higher-rank higher-spin gauge algebra and matter is fields, given etc. here by In Konstantin Alkalaev Towards higher-spin AdS JHEP04(2020)206 6 16 17 28 12 24 10 23 19 14 15 4 8 26 36 – i – 20 29 25 11 27 19 4 1 28 27 25 24 9 9 1) representations , 12 30 (2 32 8 so ) Verma modules 1) lowest-weight unitary irreducible representations 1) Verma modules R , , , (2 (2 ) representation theory massive scalars and twisted-adjoint equations (2 R sl so so 2 , (2 A.1 A.2 A.3 7.1 Covariant derivatives 7.2 Gauge field7.3 equations Massive scalar field equations sl 6.1 Twisted-like representations 6.2 Special6.3 values and factorization Twisted-adjoint action on the higher-spin algebra 5.2 Abstract definition5.3 of the twisted-adjoint Twist action automorphism5.4 and compensator vector Twisted-adjoint action5.5 on the universal Towards the enveloping algebra twisted-adjoint action module on the higher-spin algebra 4.1 Universal enveloping4.2 algebra in Howe Higher-spin dual quotient variables algebra 5.1 Adjoint action 2.2 Spectra of bulk fields/boundary operators: bi-fundamental representation 3.1 Definition 3.2 Extra factorization 2.1 Singletons: fundamental representation C Canonical basis D Twisted basis E Twisted-like B Basic star-product expressions 8 Comments on candidateA interactions 7 AdS 6 Verma modules as twisted-like representations 4 Oscillator realization 5 Module structure of the universal enveloping algebra 3 Higher-spin algebra in two dimensions Contents 1 Introduction and summary 2 Singletons in one dimension JHEP04(2020)206 ]. 11 , ]. We 10 16 , ][ λ [ 15 hs ]. However, the 12 – 10 ] defined in [ λ [ hs a real number. ) vector models in higher dimensions suggest λ N correspondence prompted by the Sachdev-Ye- ] to carry an indecomposable representation of ( 1), can be taken as a singleton module. In fact, – 1 – , 1 O 15 (2 with ] for reviews) motivates a thorough investigation of so λ 6 CFT , 2 − / 5 of 1 2 ∆ V is known [ = 2 or in the vicinity of the ultraviolet Gaussian fixed point) ∆ q which plays the role of the fundamental representation of the V is a two-dimensional BF theory for the Lie algebra ] for ∆ = is the higher-spin algebra itself, which (as any Lie algebra) carries ] (see e.g. [ 2 λ 4 [ – 1 hs ]. However, the analogue of the key group-theoretical results underly- 14 )-singlet) primary operators on the boundary. – N 12 ( O ] is already available in the metric-like formulation, as well as in the frame-like )-singlet sector. It is comforting that the HS extension of Jackiw-Teitelboim , the HS algebra is a single copy of the Lie algebra 9 1 singleton module adjoint module N – ( 7 O aaCartan`ala by a connection one-form taking valueHS in the gravity adjoint on module. AdS For instance, In two and three spacetimeso dimensions, the one gauge has sector to ofthe introduce HS bulk. a gravity is matter topological, sector in order to have local degrees of freedom in The the adjoint representation.algebra From of a symmetries holographic perspective, (whichgauged is the in a HS the global algebrasymmetries bulk). of symmetry is the the in singleton. On the In the the boundary bulk, the and boundary, “gauge” sector which it of HS is can gravity is be described defined as the Lie algebra of CFT explain that any primary field ofspanning conformal a weight Verma ∆ module together withsuch all a its Verma descendant, module the HS algebra The HS algebra. From aconformal holographic field perspective, theory (CFT) itmodule living is on the corresponds the fundamental to boundary. field a In in other the generalised words, free free the field singleton of arbitrary conformal weight. In Let us recall that, in any dimension, the spectrum of HS gravity theories is organised in Before investigating some dynamical issues in the future, a systematic study of the • • ing HS gravity theories (suchtwisted-adjoint as modules, the oscillator etc) realizations remain of to the be HS settled. algebra, the This adjoint isterms vs the of goal the of corresponding theplay higher-spin present an algebra, paper. instrumental of role: which three irreducible representations (i.e. bilinear lowest-dimensional analogues of the (by-now standard)ity kinematical theories foundation is of in HS order.already grav- begun For [ instance, the detailed study of HS asymptotic symmetries has its (e.g. in the quadratic case of the gravity [ formulation, i.e. as a BFbulk theory dual for of SYK the model two-dimensional should HS also algebra contain [ a tower of matter fields dual to the single-trace The recent surge of interestKitaev (SYK) in model AdS [ lowest-dimensional higher-spin (HS) holographicthe duality. analogies between Despite the SYK important modelthat and distinctions, (broken) HS symmetries should play an important role, at least in some degenerate lim- 1 Introduction and summary JHEP04(2020)206 n Φ AdS ] for +1. R (1.2) (1.3) (1.1) λ 21 ] is the , − λ n 20 [ exist only hs = n (where n O ) in the vector scalar fields τ ( 2 ~ ψ ..., ]) and also [ representation. The + = 19 p ~ ψ )-singlet operators Φ n N ( Φ m O Φ , irreducible mnp spacetime can be written schemat- ]. In particular, the subset of scalar . To be more precise, this matching + 1) g , 2 λ λ ) fixed by the higher-spin symmetry [ λ 2 − with conformal weight ∆ =0 1 ··· hs 1.1 − n ∞ X an infinite tower of AdS n forms an + O m,n,p )( ~ ψ n determined by λ n + ), the boundary dual of this tower of scalar n spacetime). More precisely, these scalar fields ∂ i − · 2 – 2 – 2 n M 1.1 n ~ ψ Φ 2 n ∼ = ( M n 2 n − O M 2 in fact corresponds to the propagating sector of HS grav- ) n 2 fields sitting in the twisted-adjoint module for AdS of one-dimensional boundary fields Φ R 2 ∇ N 1 ( dictionary on ( h 1 = 1 Majorana fermions (so the bilinear operators =0 ∞ d X n ) with conformal weight ∆ = 1 2 CFT N / ( are 2 = O ) with increasing masses ~ ] for an early proposal in terms of bi-local collective fields (with some similarities to ψ denote some (yet unknown) cubic coupling constants, while the ellipses λ L and HS symmetries are softly broken by quantum corrections in the bulk. ··· 18 , , 3 mnp 17 , g twisted-adjoint module are scalar fields with the above masses ( /N 2 for even (respectively, odd) integers ). At the interacting fixed point, they acquire anomalous dimensions of order one , n n n 1 , is dual to the spectrumof of the single-trace spectrum operators of onmain the AdS result boundary. of The our identification investigation. ity. The so-called “twist”sponding is intuitively an to involutive anin automorphism inversion the of on bulk. the the HSthe In boundary algebra, bulk, practice, and corre- the it to defines “matter”zero-form a a sector taking PT values twisted of in transformation action HS the of twisted-adjoint gravity module. the is HS In described HS algebra by holography, on this a sector itself. covariantly-constant In The As a corollary of our analysis, we claim that ] and See e.g. [ • form a representation of the higher-spin algebra = 0 λ 1 [ n n Nevertheless, we believe worthwhileof to SYK investigate models the with HS unbrokentheir dual HS mysterious of symmetries bulk some as dual. degenerate a small limits preliminary step inthe the previous search of bi-localdiscussions field of approach possible to HS holographic higher-dimensional duals HS of holography, the cf. SYK model [ and its generalizations. boundary fields for odd which indicates a hard breakingfor of the HS vector symmetry, models inof in contradistinction higher order with dimensions 1 what where happens the anomalous dimensions are typically bilinear in a largerepresentation number of applies only in the absence of anomalous dimensions. For instance, in the SYK model the hs stand for interaction terms ofthe higher standard order in AdS the numberfields of should fields be or a collection derivatives.This of Applying primary spectrum operators of primary operator matches the collection of resulting higher-spin invariant Lagrangian onically the AdS as where Φ forms a higher-spinstands multiplet for the corresponding curvature to radiusΦ the of the twisted-adjoint AdS module fields Φ ( JHEP04(2020)206 1) E , = 1. (2 d 1). , so ]. (2 ) represen- so 16 , R 1) covariant . It is shown , , 2 of the ]. Appendix 15 (2 (2 λ ∆ [ sl V so hs ]. These results allow λ [ realized by inhomogeneous hs dimensions as the so-called
collects various star-product d 1) B . Some technical points are fur- , 8 (2 so U contain the proofs of the propositions ], originally introduced in [ D ] realized as the quotient of the universal reviews basic facts from λ ]. We describe two types of their represen- λ [ [ λ [ A hs hs and hs – 3 – C 1) actions on , (2 1) module structure of the universal enveloping algebra , so over the ideal generated by the Casimir element can be = 1 higher-spin algebra as the symmetry algebra of a given 2) algebra and accordingly we discuss singletons in (2 d
d, so ( 1) , so (2 so , where the conformal weight can be parameterized by a new parameter U ∆ ). The resulting algebra is V , we first discuss singletons in arbitrary , the previous modules are described in terms of the , the higher-spin algebra is described using the oscillator technique and λ , we discuss , we formulate the (linearized) higher-spin dynamics on AdS , we introduce 2 6 4 5 7 3  and of the higher-spin algebra (1 ] algebra and matter fields propagating local degrees of freedom. Gauge and 1 2
λ [ 1) , hs (2 A few comments on interactions are given in section In section In section In section In section In section In section The paper is organized as follows. ] algebra. The matter sector is given by an infinite collection of massive scalars with so λ [ as ∆ = first-order differential operators in auxiliary variables. ther elaborated in thetation appendices. theory Appendix needed to discussexpressions Verma used modules. in Appendix practice.in Appendices the main text aboutbuilds two the convenient most bases general in class the of higher-spin algebra that the fields are naturallying divided of into topological gaugematter fields arising fields through belong, the gaug- hs respectively, to themasses adjoint explicitly and matching the twisted-adjoint spectrum representations of of the Flato-Fronsdal theorem. auxiliary variables used toto formulate describe the the higher-spin higher-spinning algebra algebra conformal dimensions as and antheorem thereby infinite as identify collection the the of twisted-adjoint right-hand-side representation. of Verma modules the Flato-Fronsdal with run- U tations mentioned above: thewe investigate adjoint how and two twisted-adjoint representations branch representations. with In respect to particular, the subalgebra λ the associated star-product. Weenveloping show algebra that effectively described using the star-product calculation machinery. HS algebras. Inthey the would context correspond of to higher-dimensionalthe type-A, HS Flato-Fronsdal type-B, gravity, type we etc, theorem discuss models.an which in infinite describes The which collection the main sense of tensor result Verma product modules reviewed with of here running two is conformal singletonssingleton weights. module as minimal representation of In general, one-dimensional singletons willalgebra be with defined real as conformal Verma weights modulesterms ∆. of We their review unitarity the and classification (ir)reducibility of properties, all comparing such modules with in the corresponding JHEP04(2020)206 ) , d (
. It ) or d 2) is (2.1) (2.2) 2) fields so + , 2 d, ⊕ ( (1 They also so (2) so AdS ]. In terms 2 so 2 “spin” labels, 23 C r 2)-module with , or a finite number 1) [ d = 1. d, ( ]. d, − ( so 24 of unitary irreducible so . + ∆ ). The simplest example ~s + ; 0) . ) 1) or 2 +2 R , (∆; d , 1 ( D , (2 and ∆ V − continuum )-weight made of − sl / d − d ( ( ∆ ∼ = 2). We will not use this terminology ∆ + ; 0) , so iso 2 ) (2.3) > 1) ∆ (1 2 − , λ ), which is induced from the d − + so ( ~s 2), (1  , V 1 su 2), hence their name “singleton”. (1 (∆; 1) = 1 2 − ∼ = 2)-modules are reasonable candidates for the d, V , ( – 4 – ) − d of the quadratic Casimir operator = ( R ; 0) = 2 (1 so 2 , 2 λ  so 2 ( so − m (2 ∆ d 1 4 ( = 1 dimension. The irreducible lowest-weight modules sp = D d ∼ = 2 ]. Here, ∆ is a real number corresponding to the conformal 2)-modules which do not branch: they remain irreducible m 2) ~s , d, ] and refs. therein). In practice, these irreducible modules 2) := ( (1 ) is an integral dominant 22 ). The irreducible module obtained as a quotient of the Verma d, so r d so ( so Rac( 3, singletons are traditionally defined as “minimal” (in mathemati- , . . . , s 1 s dictionary, this translates as follows: singletons correspond to bulk fields > d d ) boundary condition, satisfying ∆ ) by its maximal submodule will be denoted := ( = 1 conformal algebra is the smallest non-compact simple Lie algebra ~s − ~s d /CFT (∆; is a real parameter which, in this section, will be assumed to be non-negative is the rank of V +1 λ d r 0) without loss of generality, and Things drastically simplify in As usual in a conformal field theory, a primary field (together with its descendants) is Strictly speaking: they span two lines for the parity-invariant combination in even > 2 λ are the conformal weightsNeumann corresponding (∆ to the bulk field with eitherof Dirichelet lines (∆ for their higher-order non-unitary generalisations sometimes called “multi-linetons” for this reason. where ( role of singleton becausemodules none belong to of the them so-calledhere is discrete because series “shorter” it of or might “smaller” bringrepresentations. confusion than Up since to the normalization, it others. the containsin corresponding a the mass-like These bulk term of is such given by the eigenvalue are entirely characterised by theIn conformal weight, fact, since the there is no more any “spin” label. All lowest-weight unitary irreducible module of singleton is the conformalcorresponds scalar, to also the called irreducible “Rac”, quotient which exists in any dimension described by a (generalized) Verma module module with lowest weight [∆; weight, and where correspond to irreducible (or at most splitrestricted in to the two isometry irreducibleof subalgebras submodules) AdS when thewhose conformal dynamical algebra degrees of freedom sit on the conformal boundary [ 2.1 Singletons: fundamentalIn representation dimensions cal jargon) or “ultrashort”bounded (in energy physicist (see jargon) e.g.span unitary only [ irreducible one line in the weight space of 2 Singletons in one dimension JHEP04(2020)206 , . 2 1 4 N 3 − ∈ (2.4) (2.5) /CF T > 3 N 2 5 m = AdS ”. λ ). The results λ R , = 1 is conformal 0. In some sense, 2)-module since it + , > 2) are candidate sin- (1 , 2 . = 0) is special because so 1 as “type- (1 m ]. scalar field has negative 0 2 λ [ so 2 2)-modules leads to three > m hs , of ∆ (1 < λ < AdS 1 is exceptional because, in the ∆ . ] on the mass-square, of irreducible modules. In fact, so V 0 25 1 1 corresponds to the generic case > λ < D ⇐⇒ 6 ∆ B λ > 0 is admissible. All those Verma modules D ] for this terminology) when considered in Gaberdiel-Gopakumar λ ] where the 2 λ singletons. Accordingly, we will refer to the 31 ∼ = which is a reducible 1+ ⇐⇒ 27 of the conformal weight (of the primary field j 0 = 1 (equivalently, 0 , will be called the “type-B” theory by analogy = 1 V V , is spanned by the constants, i.e. it corresponds λ – 5 – λ 1 V 2  + 1 V = / ), but is unitary since it is above the Breitenlohner- 0 = 1 V 2.2 D  unitary = . 0 A ∆ D The unitarity region 0 0, cf ( The unitarity region V < unitary and irreducible classification of lowest-weight 2 ] and sequel) where there are two copies of 4 All lowest-weight Verma modules m The limiting case 28 ∆ V = 1 spinor singleton 6 ) are easily extracted by relaxing restrictions on weights coming from the global 1 4 R d = 1 spinor, , 1 is the usual range of values of − (2 d 6 ]). sl λ 30 6 , ] for an introduction to the representation theory of the Lie group SL(2 = 1 dimensions. They are unitary iff the conformal weight is non-negative, are irreducible and unitary with positive mass-square 26 29 d = 0 is the conformal weight of the trivial representation and ∆ λ 2 − 1+ V they can be thought ascorresponding (off-shell) theories spin- as “type-J”by (cf. analogy [ with singletons in higher dimensions where spin is quantised in general. spinor without the zero-mode, with the terminology for the(see holographic [ dual of the free spinor in higherSingletons dimensions of any spin. where only the conformal weight ∆ weight of its invariant subspace.to Physically, an a off-shell primary fielddecomposes with into ∆ the = 0 semidirectthe corresponds sum on-shell to the trivial representation. The higher-spin theory corresponding to the off-shell Freedman bound. We will refertheories to the based continuum on of those possible higher-spin free algebras unitary and singletons withSpinor 0 singleton. ∆ Holographic degeneracy. previous sense, both valuesand ∆ of its shadow) aredegeneracy” unitary. in This the corresponds AdS/CFT tomass-square the context phenomenon [ of “holographic The proof is reviewedSee in e.g. appendix [ Note that 0 • • • 4 5 3 for the Lie algebra structure of the group.only single- In or some double-valued). sense, one allowshigher-spin for holography “infinite-valued” (cf. representations [ of the group (not Accordingly, this known relevant cases: gletons in This is equivalent to the Breitenlohner-Freedman boundMoreover, [ these modules are irreducible iff the weight is positive, Unitary singletons. JHEP04(2020)206 ” ` 2)- , (2.9) (2.6) (2.7) , dual (1 s ]. so 2) is the 34 , ]. For the (1 35 and negative so related to the ), the Note that once d λ N 6 ∈ traceless symmetric N n .

) (2.8) = 3 case in [ s ; 0 ; 0 ` ` d 2; − + 1 (with − 2 2 d d . d n ] + + V V N > 2 s ) are irreducible for positive lowest- N 33 ] ( 2 ` 2)-modules span the spectrum of fields := = 1+ +∆ D 1 ) fields are nowadays called “type-A ` 37 spanned by rank- V ` d, λ , ∆ ( / =0 n ∞ V s 36 M ]) so Rac D N ]. From the bulk point of view, the modules 2 − =0 ∞ 31 1 n M , V 37 2) = – 6 – ,

= 2)-modules [ ∼ = 1 2 1 2 d, 2 36 as (unitarisable) submodule. The corresponding ; ; d, 1 ∆ ` ` ( − 2 N (respectively, Di 2 N so − + 2)-module Rac( ). 1+ , this reads [ ` ⊗ V 1 D d (respectively, Di V d, 1 2 2 ⊗ +1 − ( 2.3 d d ∆ ` 2) V so For positive integer V V d, := ` Rac( ”). The Rac ` Di 2)-modules. This type of decomposition is nowadays referred to as . We will refer to this countably-infinite collection of possible theories 0 but become reducible for even (respectively, odd) d, ) corresponds to the module spanned by bulk gauge fields of spin N ( s > so 2; is reducible, with − d N = ∆ 2 − + 1 0 s V ( E 0) is isomorphic to the D ]. n, A standard result for the tensor product between two Verma modules of To conclude, let us note that descending in two dimensions the discrete set of higher- See recent discussion of finite-dimensional singleton modules and associated topological HS models 32 − 6 ( decomposition `ala Clebsh-Gordan (see e.g. [ in [ where to conserved currents on theappearing boundary in the [ decomposition into irreducible in the bulk higher-spin gravity dual to the free conformal scalar. a “Flato-Fronsdal theorem” sincescalar it singleton was in first any obtained dimension in the 2.2 Spectra of bulkThe fields/boundary spectrum operators: of bi-fundamental boundary representation since “single-trace” singlets operators of for the vectorthe vector models representation spectra is are are extremely generated determined simple into from by bilinear irreducible the singlets. decomposition Therefore, of the tensor product of two singletons tensors (i.e. spherical harmonics). Such CFTs of type-N havedimensional been higher-spin discussed gravity in theories [ offamily A,B,C,. . of . types higher-spin coalesces theories into parameterizedweight a by of one-parametric the the continuous singleton parameter module ( The theories based on(respectively, free “type-B Rac energy lowest-energy, in which caseD the irreducible modules are actually finite-dimensional, e.g. one drops the unitaritydimensions as condition, well, one for instance has the higher-order versions of singletons in higher module quotient is the (non-unitarisable) irreducible module of finite dimension based on such finite-dimensional non-unitary representations as “type-N”. Non-unitary singletons. JHEP04(2020)206 ) – 1). 2)- 38 , 2.6 , (2 (2.14) (2 (2.11) (2.12) (2.13) (2.10) so so can take ⊕ λ ) can in fact 1) , (2 2.10 so defines the spectrum λ 2) = , , ) (2 ) has a natural interpreta- λ so  8 2.11 n ]. appearing in the decomposition . ,.... 41 2 λ . [ n ) of massive scalar modes in the , .  j 1 2 + 1)( n , D λ +1 is described by an irreducible λ 1 2.12 n − − V =0 λ 2∆+ = 0  n M N ), so that the parameter V  n =0 ∞ 3 = 1 = n M =0 ∞ The index 2) in the right-hand-side of ( 2 1 n M = 1 , = − 2 7 0 M N (2 = λ 1) = ( ). – 7 – E 2  λ , n so ∆ 1 −  ⊗ D 2.10 ⊂  n ] (cf. section ⊗ V 1 ⊗ V λ [ − 2 1) (∆ + 1 ∆ , λ N hs  n 2 V  n (2 D 1 ⊕ ) yields ] λ V = so λ = ∆ [   n 2 hs ) ∆ (1 of lowest energy =  n 1 2 3 λ m ] for a review of the group-theoretical dictionary. g ( = ) by inserting the curvature radius and taking the minus sign. Let us 31  holography as well, due to the isomorphism 1.1 2 ∆, one obtains the one-dimensional version of the Flato-Fronsdal theo- ≡ 2 /CFT as the conformal weight of the singleton, which is allowed if one forgets about 3 λ ) = ∆ 2 − 1 1 2.8 As a conclusion to this section, let us note that formula ( The spectrum of the type-N theory for the finite-dimensional “singleton” module ( = See e.g. section 4Recall in that [ the HS symmetry algebra in three dimensions is given by two copies of the HS symmetry 7 8 ]. It suggests that the integer-spaced spectrum ( − propagated by a single scalar field of mass algebra in two dimensions, in both cases. module given by the left-hand-sidewith in respect ( to the subalgebra be interpreted as labeling40 the Fourier modes inHS a theory Kaluza-Klein-like in compactification two [ of dimensions can Prokushkin be and obtained Vasiliev from in the three dimensionally reduced dimensions, HS where theory all local degrees of freedom are tion in AdS A massive scalar on AdS has been discussed in details in this previous subsection). can be obtained from thefinite-dimensional usual modules Clebsh-Gordan decomposition of the tensor product of two which reproduces ( stress that, for notational simplicity, weany will real assume value in the (positive, rest zero,∆ of or the negative). paper that Forunitarity the issues sake (in of fact, definiteness, they we will set not by default be addressed any more in the sequel since unitarity The corresponding spectrum of mass-squares reads The spectrum of conformal weights on the right-hand-side is rem ( Substituting ∆ = ∆ For ∆ JHEP04(2020)206 g ν λ R 2 ∓ 1 (re- ∼ = (3.2) (3.3) V ] and 1 4 1 . The V
λ (1) [ u 2) gl , (1 so . The annihilator g ∈ U and its shadow ] for some recent sys- ,
λ
2 47  2) 1 – , ], is traditionnaly defined 2) . Two algebras V λ , (1 [ 45 N (1 so hs / ] (3.1) ∈ so 2 λ | spanned by even polynomials of [ : it is isomorphic to the quotient C λ U | 2) by the ideal generated by the gl V ,  . ] as the quotient = ∼ ] (1
1) even 2 λ 15 )
[ ) on so 2) A ) is such that the Verma module is the quotient − λ g , (V) of endomorphisms of an irreducible 2) 2 R  hs ( 2 λ , 1 , ∼ = gl 2 (1 λ  U ⊕ V ( 1 (1 is unitary, then one will (implictly) consider (2 ) so 1 4 V R so sp V (1) , is called a primitive ideal. Due to Schur lemma, − U – 8 – u Ann( ∼ = U
(2 ] is simple for V ]. λ sl 2) ] defined in [ [ 2) = λ ] = , 2). [ λ ,
λ hs [ (1 − hs λ (1 [ -module 2 gl  . Accordingly, the singleton so d > g 1 so gl 2 V 2 -module λ ∼ = C g  ]  gl λ . The Lie algebra [ = ]. = V : gl 1 ] ] for some early seminal works and [ 16 λ λ [ 44 – gl of two-dimensional phase space. More generally, the Verma modules 2 but not for ) the universal enveloping algebra of the Lie algebra 2 42 + 1) [ , g 1) of the quadratic Casimir element ( A ν ] is isomorphic to the subalgebra ( U − 1 2 = 1 1 2 [ 2 d ) is isomorphic to the subspace of even (respectively, odd) polynomials in the λ = hs ) of an irreducible ) of the universal enveloping algebra by the primitive ideal, endowed with the ( 3 4 can be defined as the realisation of g 1 4 λ V V ( V ⊂ U admit a realisation in terms of deformed oscillators with deformation parameter Ann( ] are isomorphic iff / The exceptional isomorphism The higher-spin algebra of the singleton V 2 ) λ 2 g λ  [ ( 1 -module creation operators inside the Fockspin space algebra built from one oscillator.the Accordingly, Weyl the algebra higher- V defined by In this way, thegl higher-spin algebra define the same higher-spin algebra spectively, eigenvalue corresponding higher-spin algebra, nowadaysby often subtracting denoted the one-dimensional Abelian ideal: of the universal enveloping algebra of isomorphic to the algebra U commutator as Lie bracket. Ifthe the related real formpart of proportional the to latter theconvention algebra. in unity One in may order remove to the obtain Abelian a ideal simple algebra (this is the standard Let us denote by Ann a primitive ideal containsminus in their particular eigenvalue the on idealg generated by the Casimir operators of The modern definition ofsingleton higher-spin (see algebras e.g. is as [ tematic Lie discussions algebras of of higher-spin endomorphisms algebras of in the any dimension). 3.1 Definition 3 Higher-spin algebra in two dimensions JHEP04(2020)206 ] = ). N R and [ ], in 12 ] for (3.7) (3.8) (4.1) (3.4) (3.5) (3.6)  β hs N, T 1. The 31 48 ( , − ⊂ αβ sl 1)-vector  with 36 = , N ⊂ = +1. This is (2 J βα ) j  012 α ] into the sum  R so = 1. It satisfies − T , N 1)-invariant metric 1)-invariant metric [ , = (2 , 012  and (1 hs sl (2 αβ so  so α ) and 2.6 ) into a sum of irreducible R )-module 1 and the R , is defined by (see e.g. section 4.2 of [ N, , ( (2 N sl = 0 2 − . sl . ABC 1 )-irreps of dimension 2 )-vector index  ,  R R 1 ] , j , . AB , − 2 is the contravariant symbol, j ] is not simple and contains an infinite- ⊂ V η ). D (2 N (2 N j N ∗ [ 1 D N sl

J a, b, c, . . . N D [ αβ sp − D 2 =1 hs  )-module , j ABC  1+ M =1 N N ∞ hs  R  j M ∞ V = = ∼ = , gl M j ]. This infinite-dimensional ideal 2 are raised and lowered as follows: ) ∗ (2 , ) = – 9 –  ∼ = R ), which is finite-dimensional. The Lie algebra ] = , with = sl R 16 B ) A β R , α N 2 are raised and lowered via the = 1 N R [ N, , Y ( ), where N, ,Y 15 J 1 N, ( , hs A ( sl M B N, α ). 1)-indices read sl δ ( sl Y , N A (respectively:  gl = 0 δ (1 3.4 ∗  so − α, β, γ, . . . , N B  ” finite-dimensional δ j 1)-covariant Levi-Civita symbol M A , δ ( ) invariant metric, which is the Levi-Civita symbol (2 − R A, B, C, . . . , so ] decomposes as = (2 N )-indices are sp [ R ) is a consequence of the module isomorphisms ( , hs MNC (2  3.4 , the type-J higher spin algebra 1)-indices ⊂ +). The sp , N + +). Lorentz vector − N ABC (2  , via the − ∈ obeying to the commutation relations J denotes the “spin- so βα 9 j ,  N β D A )-modules = diag( T = R = diag( , Symplectic λ ab = 9 η (2 α AB T 1. Vector η is the identity star-(anti)commutator is denoted Following the original definition of bosonic higher-spintwo dimensions algebras in we any introduce dimension variables [ index thereby leading to the quotient ( 4 Oscillator realization The ideal where to be compared with the decomposition of the reducible Another way to understand this resultIn is fact, to one use the can principal decomposesl embedding the reducible is the type-N higher-spinisomorphism ( algebra At dimensional ideal to beis factored related out to [ the existencesome discussion). of the The submodule corresponding quotient 3.2 Extra factorization JHEP04(2020)206 (4.5) (4.6) (4.7) (4.8) (4.2) (4.3) (4.4) . B β )-invariant Y R A α , Y ). Hence, they (2 AB sp 4.3 η ), can be rewritten ] as the quotient al- = ) algebras. The basis λ in ( 4.4 ∗ [ R

, = 0. More precisely, the hs B AB β (2 ∗ ] T ,Y sp αβ , A , α ]. , t Y k ! )-singlet (  A 49 , R B AB β , . C ∂ )[ 1) and T AB −→ β are constant coefficients. Thus, we , (2 R η C ∂Y . Y . ...Z , 1 2 k
(2 Z sp B 1 A )-singlets and the universal enveloping α α (2 A 1) so ∂ Y = R ...A ←− = 0 , sp , Z 1 ∂Y ABC ∗ k A (2 αβ  αβ  (2 −  ) F AB 1) basis elements so ...A = sp , Y 1) η 1 , i.e. an ( of polynomials generated by ∗ , 3 A U – 10 – (2  of ABC αβ F 3 (2  B , t ,F  so 3 variables, ∼ = 2 1 ∈ S so S B 1 2 ], =0 ∞ run over just three values it is possible to introduce αβ β ,Z k X 3 t Z F Y

A  ⊂ A 11 S := A 1)-tensors Z α 3 ,  A ) = Y S Z (2 Z variables endowed with the Weyl star-product αβ (  = exp so A, B, . . . F Y 1 2 ∗ commute with each other, [ = αβ ∗ t which is a bi-module over

3 B β A are isomorphic [ and ,Y
A α 1) AB Y ,  T (2 αβ so  ). 1 4 U 3.2 = One can show that any element AB T In fact, this is thewhich oscillator states realization that of the the universalphic well-known enveloping as Poincar´e-Birkhoff-Witttheorem algebra vector and spaces. the Intechnique symmetric our with algebra case, the are it abstract isomor- allowsgebra definition merging ( of the the standard star-product higher-spin calculation algebra where the totally-symmetric conclude that the associativealgebra algebra equivalently as a power series in the which are, in fact,satisfy the the Hodge commutation dualized relations We notice that since indices new variables 4.1 Universal enveloping algebraLet in us Howe consider dual variables theelements, i.e. subalgebra The bilinears two algebras form the Howe dual pair is the Weyl algebra elements of the latter algebras are given by The space of polynomials in JHEP04(2020)206 ) R , (4.9) (2 (4.12) (4.16) (4.13) (4.14) (4.15) (4.10) (4.11) sl ) as the 4.7 . ). Within the . } . ) . 3.2 Z . This subspace is ) we can find that = 0 , ( B = 0 ) G , N. H ) is given by power se- n Z B.7  ) 2 ( ∂Z ] algebra elements are n ) = 0 ∂ A 1) ...A F λ [ 3 + 1) Z 4.15 ] = 4 − ( − 1) 2 ∂Z λ hs is isomorphic to the infinite- H N  − BCA ( ) ,Z ( , 1)(  AB . From (
H   3 η , [ ( − 3.2 ( AB
= BC λ T ( ∈ S + 3) = 0  1 4 ∗ , , N + 3) A H 2 = : Z  AB N Z λ T 2 , , η : 1 2 + 2)( as they are most natural in terms of the − A n A Z N + 2)( = A ( ] = 2 ∈ H – 11 – N Z G,F,G − C ( λ ...Z H ,N 2 G , µ = b D 2 { − 1 Z ∗ 2 A 2 Z ) + [ = canonical 1) is Z λ Z , 1 n n F µ 16 D (2 1 ∼ − ...A + 16 , singled out by the constraint 1 so ,Z 2 λ ] can be defined as the quotient algebra ( F A 3  C µ A λ ( [ H ) = ∂ − 0 Z  hs ≡ ∂Z act on the expansion coefficients of the power series ( > , ( n X A ] = 2 n G F  H ⊂ S Z λ 16 and its action on the universal enveloping algebra reads ) = D ∗ , / b ,N 2 D ) = Z 2 =
As a linear space, the quotient algebra ( =0  Z ∞ Z C 6 [ n M ( G N λ ) we find − = H b D and A H Z 4.11 N A Z − Using ( Let us characterize the underlying space of the higher-spin algebra. The freedom in = 2 In components, the canonical section specifiedries by the constraints ( Proposition 4.1. dimensional subspace graded by the homogeneity degree in the variables choosing representatives of the quotientsection leads to of the the problem of quotient)follows which we might basis take advantage be (i.e. of particular two most distinguishedrepresentatives bases appropriate and that the for following can proposition concrete describes oscillator computations. be approach called used In here. what Operators counting and trace creation/annihilation operations,algebra respectively. commutation They relations satisfy the where where the Casimir elementC of The higher-spin algebra oscillator realization developedequivalence in classes the previous sections the where the equivalence relation reads 4.2 Higher-spin quotient algebra JHEP04(2020)206 (5.1) (5.2) (5.3) (4.17) twisted- traceless 1)-tensors , n and (2 ), so that the so ). . are traceless but , as the ad-
4.10
) k , 1) ( 4.16 1) adjoint , , F (2 (2 = 0 ), cf. ( on itself, followed by a ) so traceless . By construction, these so ) in ( Z k
( (
Z n U ( G F U 1) 1) , ∗ , ∈  H spanned by rank- λ (2 (2 F b D , n so ∀ so ) . D m ) + U ( n U C Z ∂ D ( ⊂ ∂Z , where =0 (0) B 1) A ∞ , F M Z T m,n (2 ∗ , ] is given by rank- ∼ = ) = so ) -product. Here, elements F λ ABC k [
∗ – 12 – Z  ( ( − 1) hs F ) we obtain that the adjoint action is given by the − , F . . It relies on the trace decompositions of elements F ∗ (2 . These are defined through the = ∗ k ∗ B.7 C C
) is of the same form as ) A so A λ Z , i.e. 1) T T ( µ λ , -th copy of module U -th power µ = (2 (0) 1) algebra on the universal enveloping algebra is defined by − m k , − ∗ 2 F 1) representations that can be naturally realized on the uni- ] so 2 , (2 C ( C ,F (2 U so 0 A > so k T X [ The universal enveloping algebra denotes = ≡ F m F ). A stands for the T k ∗ -module, decomposes into an infinite collection of finite-dimensional ) ,... λ 3 1) , , µ 2 (2 , − actions of the universal enveloping algebra 1). Depending on the type of the representation, we introduce appropriate irre- that in terms of the star-product can be represented as 1 -tensors. -modules with infinite multiplicities, , 2 so , 3 C 1) 1) (2 The proof is given in appendix , , = 0 so ∈ S (2 (2 n where theso label The proof can be found in appendix Proposition 5.1. joint so Using the star-product expressions ( homogeneous differential operator The following proposition holds. 5.1 Adjoint action The adjoint action of the restriction of the actiontwo to modules the are subalgebra infinite-dimensional.of Moreover, they are highly reducibleducibility representations conditions and decomposecollection the of irreducible (twisted-)adjoint (in)finite-dimensional representation submodules. into an infinite 5 Module structure ofThere the are universal two enveloping types algebra versal of enveloping algebra adjoint and ( have no definite degree.and Thus, terms any proportional element to decomposesfactorization into is a now sum explicit, of a traceless element ( F In other words, the canonical basis of JHEP04(2020)206 . ). ∆ V + 1 5.1 ) we ) we (5.4) (5.5) (5.6) ∈ H n 5.5 5.4 (0) = 2 F A n decomposed T as in, e.g. ( D . Secondly, it
]): ∗ ∆ 1) V 31 , it follows that , , dim (2 ) it follows that the n 5.1 so Moreover, the tensor D Note that the norm of 4.17 U 12 and 11 ∈ ] in the canonical basis is a F 4.1 λ [ hs . Then, using the relation ( , 1)-modules the adjoint modules look G ). Finally, using the relation ( , n ∗ G ] are realized by the same homogeneous (2 λ D , A λ , in the right-hand-side is both lowest and highest b [ D T so n ( =0 ∞ A hs n M ∗ D T . In any case, this isomorphism is a subtle | λ ] = 0 1) on the quotient algebra, since ] = 0 = λ λ b + , D |  b D , ∆ (2 1)-module spanned by the endomorphisms of – 13 – (0) , , A V A so F T [ (2 T ⊗ A [ ] for more comments and details on this subtlety (in higher di- T so ∆ on an arbitrary element 31 V = A T F A ) have a well-defined action on the quotient algebra elements T 5.1 ) hides the need of a suitable completion of the corresponding since they decompose into the same infinite sum of irreducible ) because the trace constraint is invariant with respect to the are defined with respect to their own multiplication laws. ). From the star-product decomposition ( | 5.6 λ ∗ | 5.2 ]-modules the adjoint modules are irreducible (after removing the 4.14 ). λ [ n ). This is in agreement with the formula (cf. (4.16) in [ ) we obtain hs ∞ projected on the quotient algebra 4.17 A T ,..., is given in ( 2 ). Indeed, acting with , λ 1 b D , 10 C.10 The left-hand-side stands for the From the above discussion together with Propositions The adjoint generators ( Note that (anti)commutators evaluated with respect the star-product areNote that, labeled although by each finite-dimensional module See the last paragraph of section 2 in [ = 0 11 12 10 (1) ideal) and not isomorphic for distinct n (Anti)commutators without weight, their infinite sum isdecrease/increase neither with lowest nor highest weight (since the correspondingmensions). weights respectively module here). and can be interpreted concretely inas two ways. spanned Firstly, by the “ket-bra” left-hand-side can formed be from understood the elements of the Verma module product in the relationvector ( space in the left-hand-side,modules since involves the an corresponding infinite change linearan of combination infinite basis of relating linear the the generators. combination two has of the a generators finite may(here, norm, diverge, the even hence singleton though the each times summand completion its conjugate) of may the result tensor in a product non-unitary of module two (the unitary adjoint modules that this formula is valid for anyisomorphic ∆. for Therefore, distinct as modules, but as u one because both sides are neither lowest nor highest weight. where the line over the Verma module stands for the contragredient representation. Note see that the second term ischeck again the of first the term form defines the action of the underlying vector spacedirect sum of of the an higher-spin( infinite algebra number of finite-dimensional submodules adjoint action, see ( according to ( since where generators differential operators ( JHEP04(2020)206 . ]. τ 50 is a (5.9) (5.7) (5.8) (5.11) (5.12) (5.10) + and its g g -module, which is an associative + ) into irreducible g , there is another g g . 5.6 + is a g , − ) ⊂ g , g 2) commutation relations . In other words, ] ( ) ] of endomorphisms of the − − g d, λ g ( ( g [ y , ∈ U y gl so ⊕ − , while ∈ U g space is a symmetric Lie algebra ad τad + [ } g a, y y 13 +1 ∀ 7→ 7→ a, y d =  1) on its universal enveloping algebra, y y ∀ 1) as isotropy subalgebra. Moreover, , , g : : d, (2

− ) = , ( g ) ) ) are then y ) so g g ( so y ( ( ⊂ τ endowed with an involutive automorphism ( 5.8 y , | ] U U τ 2) of AdS g = ◦ ]. g − ◦ g ) of a symmetric Lie algebra d, – 14 – a + gl gl 51 ∈ , ( a g g ( + − isotropy subalgebra y so − → → g U { a [ . Equivalently, a symmetric Lie algebra is defined as ). For a symmetric Lie algebra ) ) a = ◦ g g g = ◦ ( ( ( g y  y U U U ). g , : : g + ) := g ad ) := is a Lie algebra a a ( τad is the Lorentz module spanned by the transvections (we avoid ( ⊂ y y 1 ] -graded in the sense that ad 2 d, + g R Z space). The relations ( τad ) on itself, via the definition , g + = ( as a canonical automorphism, which is sometimes called twist. As such g +1 [ U d τ . In both case, one obtains the same decomposition ( − transvection module g ∆ V , which will be called the with the Lorentz algebra g − g denotes the product in ⊕ ◦ symmetric Lie algebra + 1)-modules. provide a decomposition as a semidirect sum g The universal enveloping algebra A , The twisted-adjoint representation was originally introduced in the context of higher-spin theory in [ τ (2 = 13 which will be called the twisted-adjoint representation. In the mathematical literature it was defined in [ where representation of the Lie algebra is standard and defined by each other in AdS in the Lorentz basis. algebra inheriting it carries two distinct representations of itself. The adjoint representation For instance, the isometry algebra g the complement to call these infinitesimal displacements “translations” because they do not commute with of subalgebra of will be called the a Lie algebra which is we begin by discussing thisuniversal construction enveloping in algebra the U( case of any symmetric Lie algebra The eigenspaces Verma module so 5.2 Abstract definitionBefore of formulating the the twisted-adjoint twisted-adjoint action action of can be understood as spanned by elements of the algebra JHEP04(2020)206 . ) ) s a is = g g P ( ( ...A 5.4 V 2 ] = 0, (5.15) (5.16) (5.13) (5.14) (5.17) a, b A 1 ⊂ U ⊂ U A g 0) in the L, L + F , space, this g a ), where = 0, so that P -module +1 ,L A g d ) by the degree a V g , see section P ( = ( A − U g P A 1)-tensor = ( , P ), but increases by one A preserves the filtration (2 T g 2) of AdS , so | Lorentz basis elements as . ) 5.13 d, s g ( ]. The above splitting into B ( τad 1). The Lorentz subalgebra T so = 0 (for a detailed discussion, 1) Levi-Civita symbol ( , 53 , . 0 B s , cf. ( ∈ U , = V a (1 of a (irreducible) + a A P g ...A g so
2 ∀ − V A = (0 1) 1 . . 1)-covariant manner by introducing a − , 1) covariant oscillator approach devel- A , covariantized + , A ) = A (2 g F a | (2 1 V is the (2 = 1 T P so y , A ( so ad so A ◦ V = ab U  V a = A subject to the normalisation condition A + ⊂ ∓ – 15 – V g A | a V . On the other hand, V ◦ L , τ we define the τad y ,P , where preserves the canonical filtration of 5.1 ), which therefore carries an adjoint and twisted-adjoint A L B , then the latter is a well-defined automorphism of the g ) = T | V V can be represented as a rank- ab τ L ( ) :=  ( B s ad a gl τ ( T 1)-vector 2) of the isometry algebra  , ∼ = ] = y = b d, (2 1) basis elements in the Lorentz basis, , L V ,P so is the Lorentz rotation with the commutation relations [ τad a (2 of rank P L s so Ann -transversality condition, / via the definitions above applied to equivalence classes. In the case of
...a A ]. 1 we employ the compensator formalism [ g a V 1) 1) basis elements , 1) can be defined as the stability algebra of the compensator. Then, any 4 , and [ 52 , H , , b (2 (2 (2 P ). Then, the twist automorphism acts as so 48 . On the one hand, , ab ]). so so   5.8 U g 36 54 ⊂ ∈ ] = 1) 1)-tensor a  Given Finally, if a (primitive) ideal Ann The restrictions of these two representations to the isotropy subalgebra , , y 1), cf. ( (1 (1 , L, P By construction, the transvectionsthe satisfy the standard transversality choice condition Lorentz of basis. the compensator reproduces transvections so so subjected to the see e.g. [ oped in section rotation and transvections can(constant) be compensator done in an The standard representative can be chosen as [ 0 To implement this transformation within the algebra [ 5.3 Twist automorphismLet and us compensator represent vector the are transvections and preserved by the automorphism quotient representation of the singleton module Rac( construction leads to the adjoint and twisted-adjoint module of the bosonic higher-spin for in the algebra elements,by see the section degree in thethe element filtration of by the the isotropy degree subalgebra in the elements of the transvection module coincide, However, the restrictions ofdiffer these since two representations to the whole Lie algebra JHEP04(2020)206 ). (5.22) (5.20) (5.21) (5.23) (5.24) (5.25) (5.18) (5.19) C.11 ). Using along par- A 5.17 . , ) we find that , Z ⊥
|| , ∂ ∂ · 1) · 5.18 , 1) on the universal || ⊥ , + 2) (2 ∂ ∂ (2 k so . = = so N independent directions. . defined in ( A || ⊥ ∗ U respectively increase and . V  } A ⊥  ) ∈ A P ⊥ , + 1)( ∂ ,F V 5.2 F 2 k · A or and ∀ L P N Z 1) algebra commutation relations L { || ( + and , . Here, we list operators used in , , , A = C A ⊥ . − ⊥ A (2 B ⊥ = 0. Further details as well as our D P ∂ ∂ P L , Z A 1) + ( , F , so A ⊥ B , C C ⊥ ) A A Z ⊥ ). P ⊥ − V V || Z Z A P Z Z · + 2) = Z T ⊥ ≡ A · ( · + || 4.13  ∗ A ) given in ⊥ τ V N ABC ABC || ⊥ ( K Z ( A   k  ∗ ∗ Z Z K Z − · 4.12 F – 16 – ABC = = and , ] = ] = ) into account, the twisted-adjoint action can be  A = ⊥ + 3) 2 − || A 2 A B ⊥ − Z ,Z A P P T N F A , J , Z 5.15 = = ), the twisted-adjoint action of , ∗ L A V [ ∗ A L A ] P ) A [ T ) and the transverse/parallel variables ( P T + 2)( 5.12 J V · 1 2 ). We see that the Lorentz rotation acts homogeneously in = L, F ) algebra, as in ( N B.7 : ( Z R = ∗ , K 1)) is given by = [ − 5.20 ,Z A , ,Z = ( (2 T 2 ⊥ ⊥ F ,F (2 || sl A A k ∂ Z L A ∂  · T so Z · T ( , cf. ( J 1 2 − || ⊥ U ⊥ Z Z ≡ = = N 2 = = F + C k A ⊥ k T N N N = N The covariantized twisted generators satisfy the The compensator defines a covariant splitting of the auxiliary variables , while transvections act inhomogeneously since ⊥ The Casimir operator in the twisted-adjoint representation is given by where the central dot stands for an element of the universal enveloping algebra, cf. ( decrease the degree by one as discussed in the end of section Here, Z The double line brackets denote theis twisted either commutator commutator which as or follows anti-commutator,the from star-product depending the expressions on last ( line enveloping algebra or, taking the twistgiven automorphism in ( the covariantized Lorentz basis as These are counting andEach trace set forms operators its ( own 5.4 Twisted-adjoint actionFollowing on the general the definition ( universal enveloping algebra module conventions and notation canthe sequel, be found in appendix allel and transversal directions, where The two sets are transversal to each other, JHEP04(2020)206 (5.26) (5.27) (5.28) . This ) is realized . 3.2

) = 0 is spanned by of polynomials Z 3 ( 3 = 0 e S S H one will consider a H ) could be technically 1) n 4.1 A λ − − T ⊥ , k as a linear space, but we  A . Therefore, the twisted gener- k N 3 ( λ is isomorphic to the infinite- ( e Z S ) , ) = 0 for the canonical basis . This completion is important Z 3.2 ( ( ⊥ = 0 Z H . An attempt to build the projection H  λ , 1) ]. Then, one can directly see that in this case , . The deformed oscillator approach, interesting on . Another indirect argument for this is given in λ A λ − 16 λ ) = 0 for the twisted basis. However, the T ⊥ 14 ] = 0 Z 15 λ  ( b D −→ H – 17 – , . Below we just outline the general heuristic idea :( A A 1) T 3 D T [ − ) have a well-defined action on the quotient algebra ⊥ ∈ S ) fails because the subspace of traceless elements is not  ] can be explicitly built once the quotient algebra ( 5.21 H singled out by the constraint λ such that it is isomorphic to  [ 4.17 3 hs
e = S be a completion of the commutative algebra 1) ). More precisely, the commutative algebra n ). In other words, the quotient is an invariant space. Let us , 3 ) reduced on the quotient is quadratic in e W S 4.7 (2 0. It means that representatives of the quotient algebra should be ) = 0 does not possess polynomial solutions apart from the trivial e 4.14 H ⊂ 5.25 so Z 6≈ ( ] U but one allows for power series in As a linear space, the quotient algebra H  , see ( k , , Z 1) A Z n T − W 1) action both on the universal enveloping algebra and its quotients. ⊥ , is given in ( =0  ∞ (2 . n M ) on the quotient must be at most linear in λ so basis in the higher-spin algebra which is more suitable when discussing the twisted- b 15 = D 5.27 e H For our purpose, let Thus, along with the canonical basis described in the Proposition using the general formula ( The twisted-adjoint action on The twisted Casimir ( -invariant, [ 15 14 A λ A ators ( footnote in inner coordinates usingthe e.g. twist automorphism the defined deformedyields with oscillators the respect [ twisted-adjoint to action thein linear associative its in product own, the in will parameter the be deformed considered oscillator elsewhere. algebra The detailed proof is givenbehind in the appendix Proposition. Proposition 5.2. dimensional subspace subspace is graded by the homogeneity degree in the variables by the modified traceequation condition ( ( one. Strictly speaking,enveloping algebra one shouldwill introduce keep an this analogous subtlety implicit completion in of the sequel. the universal adjoint in the variables polynomials in because we propose to replace the trace condition chosen in some othercumbersome way. because using Doing the the canonicalanalyzing explicit basis the projection in twisted-adjoint the representation. to higher-spin find algebra is inadequate for twisted where the resulting generators dependT linearly T The twisted-adjoint generators ( elements since where denote the projection of the twisted-adjoint generators on the quotient algebra as 5.5 Towards the twisted-adjoint action on the higher-spin algebra JHEP04(2020)206 , k k k n = − of 16 d Z was W = 2 ... n ⊥ 3 ) = 0 k ... e  (5.29) (5.30) S ⊕ d Z 2 ( ⊕ ) H 2 1)-module 1 d ,  ⊕ D (1 1 1)-module ⊕ , 1 so N d (1 of all ranks ( ⊕ D = 1 or dim so of each irreducible ⊕ . 0 n 0 ..., ) 0 D D d n • • • • • • • • • • d

⊕

d • • • • • • • ... • • •

=

2 • • • • • • • • • • ∼ = ⊕ d ⊕ H 2 k d ... is the trace creation operator W ), we arrive at the one-to-one ⊕ 2 ⊥ ⊕ ⊕ ∞ · 1 Z 2 1 1)-modules d d 5.30 , d • •

⊕

• (2 • ⊕

0

are shown in red contours. The counting • • 1 d so n ( d ⊕ ∞ · is singled out by the constraint ( W (of dimensions dim ) and ( is singled out by the relation ⊕ 0 3 k ∞ · d 3 e 0 S d N ), the dependence of the parallel coordinate d 5.29 k = + and its square ∞ · ∼ = ,Z N – 18 – ⊥ ... BBB ] in the twisted form. Each Young diagram of length BIG e ⊥ = H ⊂ S ) = 0 can be represented as (see figure have a well-defined action in each subspace H ⊂ n λ Z [ Z ⊕ Z 1)-module structure. In the next section, we will D ( . Subspaces ( A λ , 2 hs ... k T H d H (2 ⊕ 1)-modules 1) so 2 , ⊕ W ) = Baas . 0am − 1 (1 Z traceless Lorentz tensors, leads to ⊥ ( ⊕ D so 1 k  H ⊕ W 0 BBB . Now, comparing ( BBB ⊕ D k 0 ]. decomposes into the infinite sequence ①BM 56 BBB 1)-covariant basis of e k irreducible module H ≡ W , ] and a functional class in oscillator variables analogous to the completion 0am BBB k W (1 H ≡ D 58 - – so 55 has a fixed eigenvalue on all elements inside a given contour. direction). Since k 1) rank- ) or, equivalently, in terms of “spherical harmonics”: ). Now, the branching rule . The , N 0) spanned by rank- ⊥ (1 ) = 0 which is the two-dimensional Klein-Gordon equation and its general solution ,... 1)-modules as a sum of 1)-module 1)-module for any Z 4.15 so 2 The twisted-adjoint generators In the twisted basis, the subspace In the canonical basis, the subspace , , , ( k > In the context of the continuous-spin field theory, such a modified trace condition was previously , is not constrained by the Klein-Gordon equation so that the homogeneity degree in has infinite multiplicity. As a linear space, each infinite-dimensional indecomposable 1 (2 (1 (2 H 16 k k , thereby endowing them with an considered in [ previously discussed in [ Here, again, the infinite coefficientd on the right-hand-side meansso that each so correspondence between the canonical and twisted bases realizing the quotient space. Z simply defines an extrasolutions index to running the from constraint zero ( to infinity. In other words, the space of 1) has the form of anical infinite harmonics” expansion on in the powers of circle,In the i.e. our radius Fourier case, modes, with the coefficients which analogue being of can(in the “spher- be the radius written is in covariant form. where the infinite coefficientsenters in in a the (countably) infinite right-hand-side number mean of that copies. each which is the three-dimensionalgiven d’Alembert in equation. terms of The0 the solution irreducible space finite-dimensional issee known ( to be so for Figure 1 is an operator JHEP04(2020)206 ]), are with 54 (6.2) (6.1) n (5.31) n W 1) con- ∆ , V (2 1) genera- representa- , so ) of the only t (2 ( , φ so N N A ⊥ conformal-weight ,.... ∂ 2 resembles the way  -variables, . Subspaces , ) , k 3 1 ⊥ k • • • • • d ,

S • • Z • • • • • • ∆ • • N

twisted-like • • • ( • • • • • • •

− • • • • • = 0 ∆ d dt + . These are always infinite- t
⊥ ) are satisfied for any function − 1) N , , n =  5.24 has a fixed eigenvalue on all elements • (2

− • • k

+ 1 so • • A ⊥

N 1)-tensor module • λ Z , U are shown in green vertical contours. The − (1 ,D 1) referred to as = n , n so are known as Weyl modules (see e.g. [ ∆ A D N t (2 N P = 1) on the linear space n + + , ) are isomorphic to Verma modules so ) can be realized, on functions n ] in the canonical and twisted forms. Each Young belong to the general class of W + 2 , N λ (2 – 19 – N BIG [ λ A BED d so dt 5.28 hs P traceless 2 t . It will be referred further as the k k in ( P,K,D 1)-modules = , , and N n B ⊥ (2 . ago L ago ∂ W so A ⊥ has a fixed eigenvalue on all elements inside a given green contour. Z k ,K C N , as inhomogeneous differential operators BBB factorization R V + d dt ①BM ⊥ ∈ = N t with the weight ∆ ABC BBB 1)-modules 1)-covariant basis of  stands for a rank- ①BB P = , , . − k (1 BBB (2 N DBM - = n so for details. The commutation relations ( so - ∆ L E V . 1) on its universal enveloping algebra , ∼ = 7 ) such that transvections are realized inhomogeneously in . The (2 n . so W ) which is a power series in 5.24 k Note that the above realization of N ( the conformal algebra actsformal on algebra primary with fields basis of elements“spacetime” ( some variable CFT. In our case, the see appendix ∆ function 6.1 Twisted-like representations The twisted-adjoint generators tors ( We introduce a classtions. of representations Such of representationsgebra generalize the twisted-adjointdimensional modules representation decomposed into of an infinite theconformal collection weights. of Lie Verma modules al- with running In the higher-spin literature,cf. the section modules 6 Verma modules as twisted-like representations show that the conformal weights depending linearly on Figure 2 diagram of length shown in red horizontal contours.inside The a counting given operator redcounting operator contour. The JHEP04(2020)206 ) 0 L 6.1 (6.9) (6.6) (6.7) (6.8) (6.3) (6.4) (6.5) ) = 0 Z ( . ] can be ) algebra H λ [ R 1) ), it follows , 1) hs − (2 6.4 − ⊥ sl ⊥   ( A ⊥ ) and, therefore, one ∂ . || ), i.e. homogeneous . From ( } , k ,Z N 6.1 ⊥ is sliced according to the , thereby endowing it with 1] = 2 is then bi-graded as follows m F Z + ∆ 1)-module specified by the e ( H 3 , − is given by ∆ = F S N (2 is always an infinite direct sum n ⊥ H F = ∆ so  ⊥ ∆ ) = , V 0 , H A Z ). Introducing the counting operator ( . P , 1) [ . A F 1 : n we find the isomorphic − n,m ∆ n F , N , − ,L n V U 1)-module D 1] = 0 = ⊥ , 1) (∆ ). Thus, − =0 =0  ∞ − (2 F n n n M − ∞ k M = 1] = 0 ⊥ so n,m – 20 – = ∆( N 0  − and = ∆ , = : L ∆ ≡ || considered as an || ⊥ + 2∆ n 3 , ] 3 n H N N  S 2 [ e , N K H C ( L [ t ∈ S [ ). The  singled out by the modified constraint ( k , where each submodule is isomorphic to the Verma mod- k  F 3 N P , the linear space of the higher-spin algebra , e ( N S can be represented as d dt { ∀ = ∆ 3 5.2 = =  e ] = 0 H ⊂ ||  ∈ S L . L  n,m F ,N ). U L A degree, respectively. The linear space of P || [ E.18 ), any , of the operator be the vector space are subspaces of fixed degrees, n ∆ 5.18 and defining ] = 0 ) and ( H with the conformal weight ∆ 1)-module structure. n,m || d , dt U n degree and t (2 ,N ∆ E.17 Let The important property of the twisted-like representations is that the generators ( ⊥ L V so [ = where 6.2 Special valuesBy and using factorization ( can introduce filtrations with respectare to degrees in transversal and parallel variables which The Casimir operator eigenvalue on the Verma module see ( ule of Verma modules with weights defined by the particular conformal-weight function, and further sliced according tothat the counting the operator twisted-like eigenvalues generators havean a well-defined action on conformal-weight function eigenvalues Moreover, these two operators trivially commute with each other, Following the Proposition realized as the subspace weakly commute with the operators N basis elements which are schematically the sameand as inhomogeneous the twisted-like generators ( where ∆ is a conformal dimension (see also appendix JHEP04(2020)206 . || ), 0 6.11 (6.14) (6.15) (6.16) (6.10) (6.11) (6.12) (6.13) ⊂ Q l degrees. Q ⊥ degree not keeps the 1)-invariant), A ⊥ , P (1 defined in ( ), hence subspaces so will be referred to A − of the acts by lowering the W m . 6.12 l A − 0 is invariant with respect Q . W 0 A ⊥ Q ∂ ⊂ Q  , invariant subspaces provided 1 1)-modules. For the sake of ) l implying that we have chosen , k − A , , N (2 m ( ⊂ Q degree, P 1 so +1 − ∆ ⊥ ] = 0, and, therefore, ... , ≡ U k , + n,m n,m ⊂ A − . ⊥ . ) isomorphic to the universal enveloping ,N n,m 1 0 l n,m W N A − U  → U → U l 6.8 P + − → U ) the whole space ⊂ Q ⊂ Q A + l n,m n,m 0 ⊂ Q = W 1)-invariant. U U – 21 – Q l . Thereafter, the number Q n,m 6.12 , A − A = n A U Q labels different (1 P W defined by an arbitrary conformal-weight function P A : : : k by raising the grading number ( so P . In contrast, the operator A ) we have [ L N 0 0 A + P A − is , N Q W , 6.4 W of m A ∈ ⊥ U n,m n l Z l U = ∞ = acts in M m A + implicitly and denote A + = W l n W ) does not have this property, mixing up terms of different ) and its natural invariant subspaces are the complements to Q may be an invariant subspace for some ) it follows that each level is Lorentz-invariant (i.e. : respectively increase/decrease the . First, from ( 0 6.10 0
1)-module with fixed A 6.12  , N degree remains intact under the action of the twisted-like generators we 1) 6.13 , (2 W ⊂ Q ), which act on the vector space ( ∈ || l l (2 so invariant at any 6.1 Q − so A + U W . From ( ), cf ( k In general, by virtue of the property ( Now, let us define the following filtration by the subspaces Since the Let us consider transvections are N level ( l Note that the operator Q grading number ( Then, considering their sum we find out that there are However, for special values of thea prefactor subspace in the decreasing operator to the action of the transvection a particular as while transvections shift levels back and forth. greater than and, therefore, any subspace conclude that the eigenvalue simplicity, let us keep though their sum ( The Lorentz rotation does not change any of the two degrees, where The operators ∆ algebra degree invariant. Second, the transvectionsinto are two inhomogeneous parts operators as that can be split JHEP04(2020)206 . n on For ). ⊥ − ∂ (6.19) (6.20) (6.17) (6.18) (6.21) λ 2.6 17 1. = − m by the modified 1), both in the is the irreducible , λ , . . . , λ l = 0. The resulting 2 S (1 D , n , there are only two acts by zero); 1 so and of the conformal- ⊥ , ⊥ )). N ), we conclude that the ⊥ ∂ with . N = 0 0 we arrive at the module . 0 0 6.17 D , n D.22 Q ≈ N 0 ). The factorization yields the 1 + 1. , ∈ l ≈ 1 − λ ) − on which 6.19 n λ ⊥ ) has zeros at the levels ( .... ⊥  for ∆ Z l ⊕ ⊕ D 6.17 Q +2 / 1 + λ 0 ... , ), see appendix : Q − because the zero arises at level 1, cf. ( ⊕ 0 l 2 0 ⊕ D + 1 = Q 6.20 – 22 – D 1 ) of dimension 2 λ +1 − l ⊕ D λ − , but in this case the zero is achieved by acting with 2.6 means that it holds when acting on polynomials 1 is reducible because there remains the modified trace D ( λ k l l the equation ( N ⊕ D = ≈ D D = 0 : ⊕ D λ n n . As the prefactor is linear in l 0 n D ) = Q 1)-module k D , 1)-module. This is essentially the same mechanism of quoti- A − ). N , (these are constants in on the left-hand side. On the other hand, the right-hand ( (where the prefactor equals zero ( (2 W (2 0 . 0 ∆ 3 vanishes for particular eigenvalues of so 6.17 degree so 1)-module A 1) module A − , labeled by , || ⊂ Q ⊂ Q W (2 0 ), 0 l (2 n U U Q so with eigenvalue zero on so ( A − ∆ W = 1 so that there is just one zero in the module is the trivial λ 0 ) and in the resulting quotient ( ) with fixed D ) specified by the conformal-weight function ( || 6.21 ,Z the subspace the subspace 6.6 ( Further imposing the modified trace constraint In general, the By way of illustration, let us consider e.g. the conformal-weight function ⊥ Formally, one should also consider − − Z ∆ 17 ( This can beideal directly ( seen by counting elements inconstants the as discussed basis below ( of which is shownside on is figure the infinite-dimensionaltrace module constraint, singled out from the subspace instance, fix quotient H finite-dimensional module Then, for modules It follows that the quotient spaces arises in subspaces with constraint to be imposed.subspace singled out Following by the thefinite-dimensional discussion modified below trace ( constraint in the quotient As a consequence, the quotient is a (finite-dimensional) enting with respect to invariantas subspaces as reviewed for in Verma appendix modules and singular submodules, weight function where the weakF equality symbol eigenspaces of that the prefactor in JHEP04(2020)206 ). (the ). In 3.4 (6.22) λ as the ] and the 3.4 modules,
λ [ n 1) hs , ∆ . V ) is quadratic (2 A ⊥ 1) algebra, i.e. ∂ , N N so
and 4.14 (2 l ] in terms of inner U so λ + 1 D • • [ • • •

) with the conformal- • • • • • • • λ • • •

hs • • • • • • • • • •

.

− • • • • • 6.1 A λ k T N 1) with generators acting as , ) that exactly corresponds to + (2 ⊥ ]. 3.4 so N • 11

18 [ • •

in line with the factorization ( ). • • λ

− ] of • ] to obtain Z λ ). As an example we can examine the 3.8 A [ ⊥ λ ) and using the factorization properties [ ∈ k Z hs hs N λ 6.19 N = ( N + A + ∆ λ agrees with ( , we can fix uniquely how the twisted-adjoint ) and ( both the quotient and the subspace are given P ) at integer N N BIG R – 23 – thereby guaranteeing the isomorphism ( 3.6 BED N 3.2 6.20 ; ∼ = , one of which yields the higher-spin algebra λ ). Green and red contours show
∈ λ 1 , 1) J 6.6 λ , B ⊥ / ∂ (2 ) has zeros at A . ago [1] ⊥ ago so k Z gl N , see ( U ( C singled out by the modified trace condition is sliced into Verma 1) generators acting on the representatives are linear in V , ∆ ,... ] at integer we do not explicitly build the quotient ∆ BBB 2 factorization λ (2 can be projected onto , ) is linear in ①BM [ ) is exactly given by ( H k 1 k
1 + ABC so , 5.5 hs  N N ) satisfying the following conditions: 1) BBB ( − ( k − , ①BB l , reviewed in section = 0 . ∆ N ) in accordance with ( ∆ λ (2 = ( BBB n 1)-module defined by some DBM - λ so , ∆ - L 6.21 ). At general integer (2 U with so n on 6.18 ∆ : . The subspace ). V A ) and ( T ] at integer A λ λ T 6.19 [ The prefactor The function and, therefore, the Finally, we obtain that the solution is uniquely given by The condition (a) follows from the fact that the equivalence relation ( In order to build the twisted-adjoint module 6.20 from ( On the other hand, this factorization can be shown by analyzing two types of the twist-invariant ideals , we proceed as follows. Let us consider a twisted-like action of the . The resulting hs λ = 1 case. Then the quotient 0 18 k (a) (b) A λ in the universal envelopingother algebra one guarantees the further factorization ( fact, this last conditionN uniquely fixes theλ conformal-weight function to beD linear also in by ( in Lorentz rotation arethat intact, the the twisted-module transvections structureof are on additional modified). the ideals equivalence in classes The must condition conform (b) the existence claims weight function of action T the Lorentz rotation and transvection generators are realized by ( 6.3 Twisted-adjoint action onAs the discussed in higher-spin section algebra coordinates. Instead, startingtwisted-like from the universal enveloping algebra Figure 3 modules respectively. The verticaltion ( violet line visualizes the factorization for the conformal-weight func- JHEP04(2020)206 e 1) , 0 (7.5) (7.3) (7.4) (7.1) (7.2) , ) and (6.23) taking 2 5.16 = (0 M A ) can be read satisfying the and Λ are the V 1). To describe 1)-tensors. R , , W 5.28 (2 = 0. Note that in = 0 so A e m , ( A V A m m ,..., , . The associated curvature x 2 W 1 , , m ]. ). m A 1 , A Λ = 0, where 9 ω 1)-valued connection one-form , – , where dx C − 7 1) with arbitrary ranks in this W 2.12 (2 , ω A W = = 0 R ...Z V . Then, all derivative objects (like (2 so is the exterior product. The metric ∧ 1 + A B n A so = e e B ∧ W Z , A m = W ) 1)-vector field, e -form fields on the manifold , x , n -forms and explicitly p ( W p AB (1 n ABC η ) = 0 2). Using the compensator vector (  so , ...A + 1) 1 2 1 ω , ω 1 = we have a decomposition W , – 24 – ( λ A A where 3 + A e U is invertible. Note that the above three equations S V − and spin connection mn A R = 0 g n a m ⊗ − =0 ∞ a m e X n A ) )( e , A as the sum dW 2 λ A ω W = T ) = − M ( A Z = A n p | R Ω x A ) the connection can be split into the frame (zweibein) field W e = ( 2 matrix ∈ U( = n ) ] × 5.17 2 Z | W C [ x be the space of differential is the de Rham differential, and 1) basis elements ( 3 m , e S ∂ spacetime is described locally by the connection one-form (2 ) and takes values m ⊗ 2 so ) massive scalars and twisted-adjoint equations dx 2 E.17 2 be a two-dimensional manifold with local coordinates = are ), consistently with conformal weights given by ( M ( 2 d A p T The AdS M 2.13 . For arbitrary U( A Z where the expansion coefficients are implicitly 7.1 Covariant derivatives Let Ω values in thefibre. totally-symmetric These representations can of be packed into generating functions in the auxiliary commuting variables zero-curvature condition provided the zweibein 2 are equivalent to the singlestandard Jackiw-Teitelboim equation scalar curvature and the cosmological constant [ where is defined in thecurvatures, standard covariant derivatives, fashion etc) as can be introduced in the standard way. such that the framehigher field dimensions is the a Lorentz covariantized dimensions rotations it are can given be bywe antisymmetric dualized arrive matrices, to at but give the in a standardtwo-form two vector. is zweibein given Fixing by the compensator as and the covariantized basis ( and the Lorentz spin connection 7 AdS Let gravitational fields in two dimensions, we introduce an off from ( cf. ( The respective Casimir operator eigenvalue evaluated on the constraints ( JHEP04(2020)206 ), of s 6.1 (7.9) (7.6) (7.7) (7.8) D (7.12) (7.10) (7.11) ) singled out by ) as the covariant ). Z . | 7.6 5.2 x ). Therefore, they de- = 0 s 7.4 , ...A , the solution is given by a ) singled out by the modified 3 s . A Z A 2 | A A x 4.1 T 1 , ( A A C ...Z R m 1 = Ω A , spacetime ( , , = 2 Z A A C 2 A e ) ) Ω = 0 ∇ 1 T T x = 0 A e A A A ∇ ( 1) in the twisted-like representation ( T s , C A W W (2 ...A 1) W + 1 + so A – 25 – d , η d − , + ) Ω s ⊥ d A = = ( T  =0 ∞ e ( ...A s ∇ 1) in the adjoint representation ( A X ∇ ≡ 1 , R A ( Ω (2 ) = = m ∇ so Z | x = Ω ∇∇ s Ω( ]. ...A Ω = 0. Following the Proposition 1 λ [ A  ]. There are no local physical degrees of freedom propagated by the hs m ) are basis elements of be the subspace of differential one-forms Ω = Ω( Ω L 11 1), , , be the space of differential 0-forms ]). , A e 10 (2 59 H P ⊗ H ][ ⊗ so ) ) λ 2 [ = ( 2 are basis elements of ) are covariant constancy conditions in finite-dimensional representations (see e.g. hs A M A M ( T T ( 1 0 7.11 The equations of motion are written using the adjoint derivative ( Both derivatives being squared yield the curvature 2-from We can also introduce a class of covariant derivatives that can be called twisted-like The adjoint derivative reads 7.3 Massive scalar fieldLet Ω equations trace constraint linearization of the fullalgebra equations of motion followinggauge from connections the because BF the actiontion with ( action the is gauge topologicaldiscussion while in the [ resulting equations of mo- constancy condition where the connection one-formscribe describes the the linearized AdS dynamics in the gauge sector. These equations can be obtained as where each expansion coefficient is a one-form with totally-symmetric and traceless indices, They are the gaugehigher-spin algebra fields in the adjoint representation arising through the gauging the collection of 1-forms takingthe values algebra in finite-dimensional irreducible representations 7.2 Gauge field equations Let Ω the trace constraint where defined by particular conformal-weight function. where derivatives as JHEP04(2020)206 ) ) n x ( W ) as 7.4 ) , and n (7.17) (7.13) (7.14) (7.15) (7.16) C ( 7.7 ≡ AdS R n , } ]). The resulting , ,... 54 2 ). Then, the sys- , 1 + 1) , λ 6.23 = 0 − scalar fields Φ . 1) in the twisted-adjoint 2 AdS k n , 2 R | )( (2 = 0 ) 1)-tensor field, λ k , x so ( − (1 ...a ), , ,.... k 1 n 2 a so ( is given by ( ) , ...a n ) we can project the original equa- 1 1 C ( 7.14 = 0 n = a , degree, ) 2 The resulting equations in each irre- 2 n W n C a C ( k 7.14 1 spacetime of curvature radius
= 0 { a M A 2 λ ,... = , T 2 n A , n 1 W , W subjected to the zero-curvature condition ( , η , n – 26 – ) + A k d = 0 W = 0 ...a are basis elements of 1 n ]. according to ( a ≡ A C λ ( λ ) [ T ) e n H C C ( n hs where with masses λ with ⊗ = e ∇ − k ) n ]. From the analysis of the previous section it follows that k such a collection of fields form the twisted-adjoint represen- 2 ...a N W , , 1 ( 61 M a , n ) 5.2 ( n 0 C ( 60 W = 0 Ω In two dimensions, the only propagating degrees of freedom are in n =0 ∈ ∞ Φ ). n M 19 spacetime, and
C 2 = n 2 6.22 e M H . + ⊗ 2 ) is the d’Alembert operator on the AdS 2 ,... ) onto subspaces ) describes physically an infinite tower of on-shell AdS ∇ 2 2 , M 1 ∇ ( 7.16 , 0 7.16 Decomposing The equations of motion can be written using the twisted-adjoint derivative ( Ω In the literature, this is known as the central on-mass-shell theorem that explains how the two types of = 0 19 twisted-adjoint equations form adegrees coupled of system freedom. of equations that describes propagating equations, for 0-form and 1-formphysical HS degrees fields, of are freedom gluedaltogether to correspond each form to other a propagating (for HS massless reviews, multiplet. HS see e.g. fields [ and a massive scalar field which n 8 Comments on candidateIn interactions higher-dimensional HS gravity theory, already at linear level, both the adjoint and the given by the lowest component in each Weyl module ( where tion ( ducible sector are knownare as the the Weyl unfolded modules equationsthe [ for Casimir scalar operator fields, eigenvalues on while eachtem subspaces subspace ( where again the connection one-forms describe the AdS representation ( Following the Proposition tation of the higher-spin algebra the covariant constancy condition where each element is a totally-symmetric and traceless The two conditions above candimensional be spaces solved labelled in by Lorentz the components eigenvalue as a collection of infinite- which may be even further restricted by fixing the JHEP04(2020)206 ]. ). 20 ]). ]. 11 1.2 , (A.1) 67 be an 67 , 10 ) of Lie α α V 26 V ( ). We ignored gl 1.2 → ) R , ] BF action [ λ (2 [ H. sl . The action of the Lie hs : ] = α ,... π 2 E,F , v [ 1 ) satisfying the Chevalley-Serre R , v , 0 v (2 sl F, 2 − is a morphism ) representation theory (see e.g. [ R α , ] = V (2 – 27 – sl H,F [ ) define the quadratic part of the Lagrangian ( 7.17 ) can be explicitly constructed as follows. Let ]. Such a theory inevitably involves dynamical matter R , λ E, [ (2 hs sl ] = +2 ]. H,E 66 [ – 62 , ) on the representation space ) Verma modules be basis elements of the Lie algebra R ) representation theory 48 R , , R (2 , ). At the level of equations of motion, a non-linear theory could be built by (2 21 sl (2 sl 1.2 sl E,F,H The Verma modules of X.B. is grateful to LPI Moscow for hospitality in summer 2018 where this project was The massive scalar equations ( Strictly speaking, integrating out fields with local physical degrees of freedomIn generically this subsection, leads one to will follow the standard conventions from mathematical textbooks, e.g. [ 20 21 algebra nonlocalities. Although the gaugenot obvious fields that which the corresponding arethis effective integrated Lagrangian subtlety out would in be are the of topological above the in simple argument local for the form the present ( sake case, of it simplicity. is relations infinite-dimensional vector space with basis elements Here we recap some basic facts from the A.1 Let supported by the RFBR grant Noof 18-02-01024 Theoretical and Physics by the and Foundation Mathematics for the “BASIS”. Advancement A A. Latyshev, K. Mkrtchyan, C. Peng, D.A. Ponomarev, Yan, A. J. Sharapov, Yoon, K. for Susuki, E. useful Skvortsov, discussions. initiated. K.A. is grateful toin AEI Potsdam winter and 2019 IDP where University of afor Tours for hospitality part their in of hospitality October this 2019 work where was those done. results were We announced. acknowledge The the work APCTP of Pohang K.A. was Acknowledgments We are grateful to N. Boulanger, A. Campoleoni, M. Grigoriev, D. Grumiller, C. Iazeolla, interacting through the topological gaugeIntegrating sector out the described topological by degrees the of of freedom, the would matter leave only sector,form infinitely many governed ( scalars by aemploying higher-spin the rich invariant algebraic Lagrangian structures of underlyingdimensions higher-spin the [ interactions known schematic in higher sector (topological) and the matter sectorand (dynamical) twisted-adjoint described equations. respectively by the adjoint It is an interestingby and the important higher-spin problem algebra to build a full interacting theory controlled the matter sector so the linear system decouples into two independent sectors: the gauge JHEP04(2020)206 0 0 n v v 1)- n> , v (A.9) (A.6) (A.7) (A.8) (A.2) (A.3) (A.4) (A.5) 1 is a (2 ). The − R so , N (2 sl = ∼ = α 2, and the finite- 1) and the commuta- , α/ } (2 − will be identified with . The negativity of the +1 so D. , α being its weight. All N ,L span an invariant infinite- d P, 0 dx C of negative highest-weight. ] = 2 x 1 ∈ D. ,L 7→ − , implying that there are two 1 2 ,... − ], except for the sign of the quadratic , α n − N 1 +1 26 +1 1), it is useful to take inspiration = − P,K L − [ , { N V n . It follows that if 7→ − , v (2 n , v 1 v , − so N n = 0 this is the highest-weight vector v N v + 1) is given by the commutation relations − ) ,D n D,L K, n n } ) of conformal weight ∆. The isomorphism 2 d , − /V dx − R 1 2 P,H − ) with highest weight α +1 7→ − − ): at ∈ x ( ] = n R α N 0 )-module of dimension – 28 – E 7→ , v n − x ( V R P,D,K α are consistent with the identification (2 , = = ( = = { π = sl D,K n (2 n n n [ + v v v N sl of ) ) ) m (3). For this reason, one will make use of quantum- D F E α ) will be identified with the finite-dimensional H L ( ( K,L ( ) so V ), then the vectors α α K,F α n ,K 1) with lowest conformal weight ∆ = N π π π A.5 , P, is then identified with a primary field, while the vectors − d 7→ ∈ (2 7→ − dx 0 m 1 v E ] = so N = is the highest-weight vector with − The generators will be denoted ) on the real line ( L of 0 P x v ] = ( ( D,P 22 ∆ n [ φ V )-module ( 0 guarantees that there are no further invariant subspaces. ,L R , m < 1) is achieved by the identification ) we see that the prefactor is quadratic in L (2 , + 1 there is possibly a singular vector 1 . Here, 1) in the conformal basis sl 0 1) lowest-weight unitary irreducible representations 1) Verma modules (2 , α ). The vector − , , A.4 N (2 so = (2 (2 N 2.6 ∈ so ∼ = n − n so so ) R From ( , We closely follow the conventions and notations from the review [ 22 (2 mechanics notations. tion relations [ Casimir operator and for the symbol of the conformal weight. are the descendants. A.3 To discuss the unitary irreduciblefrom representations of the textbook example of Accordingly, the Verma module the Verma module dimensional module ( It can be realized by the vector fields acting on functions sl A.2 Let us now rewrite the abovealgebra construction for the conformal algebra The quotient space is the irreducible finite-dimensional weight are linearly independent by construction. vectors which could be annihilatedand by at non-negative integer (i.e. dimensional subspace isomorphic to the Verma module where algebras defined by JHEP04(2020)206 are real (B.3) (B.1) (B.2) 0 . One (A.13) (A.10) (A.11) (A.12) L d dx n − with eigen- 1 0 x , , ) ) L = Y Y , . ( ( n F F L ∆)   − 1 + ∆) βB βB , , . i i ) − + 1 1 2 2 ∂Y ∂Y +1 ∂ ∂ − + 1 m m αA αA L 1 m m . | | ∆)( − ∂Y ∂Y . In the doublet variables we get that appear in the representa- A L α ) as they define the left (or right) − + ∆)( A ∆) ∂ is an eigenvector of irreducible module must be a αβ αβ + Z ∂Y   m i m m − AB 1 B T 1 + ∆) m + + − | α ∗ L ) to be non-negative is that the lowest Y − + 1 the corresponding eigenvalues of AB AB F +1 − m m unitary L L ∆) = ( ∆) = ( L ∆ 2 2 A.12 ( B α − − V (or 1 2 ∂ − + ∆)( – 29 – ∂Y B B impose that the right-hand-sides of the above F , − − + ∆)( i β β A ∗ 2 0 n Y Y α m m m L | − ) and ( Y ( ( A A AB α α L = m p p = 1) + ∆(1 T Y Y
− − − with lowest conformal weight ∆, the value of the Casimir A.11 = − + 1) + ∆(1 αβ αβ 2)   AB ∆ = = = , 4 4 m m † n and in singlet variables L V and one can check that the necessary and sufficient condi- i i i ( (   (1 L 1). Below we explicitly calculate such expressions, both in A 1 8 1 8 m m m m m α 1). Using the commutation relations and the expression for the , | | | so ,... Y 0 1 − = = (2 = 2 ,... +1 − L C i i ) = L L so Y m m AB ∆ + 2 | | ( T 1 , F ∆ + 2 − +1 ∗ ∗ , ) L L 1 Y +1 − ( AB ∆ + 1 L L T F | | , ∆ + 1 m m . , h h = ∆ m − m 0 is given by = ∆ The Casimir operator is equal to > m where action of theoriginal algebra doublet variables the left and right actions with B Basic star-product expressions We are interested in products like ∆ tion. For− a lowest-weight Verma moduletion for the right-hand-sideconformal of weight ∆ ( isabsorbed non-negative. in the Up definition to of some the arbitrary basis phase vectors, factors one which finds can that be the orthonormal basis for Unitarity and theequations relation must be non-negative for all values of The eigenvalue of thenumber. Casimir On a operator Verma on module operator a is equal to ∆(∆ Casimir operator, one finds that considers an orthonormal basisvalue where the element which corresponds to their realization as vector fields on the real line JHEP04(2020)206 ) into B.2 (B.4) (B.5) (B.6) (B.7) (C.6) (C.1) (C.2) (C.3) (C.4) (C.5) n in the F n ) and ( of arbitrary , . 3 the same logic ) ) B.1 1 Z Z , ( ( − ) ∈ S . n F F 2 Z ( G   G − is of maximal degree n C C F 1 F ∂ ∂ C − ∂Z ∂Z n ∂ 2) ∂Z B B H − Z Z  n , . K 1 . , ∂ ABC ABC = ( − ). Further decomposing   = 0 n A n ∂Z 2 , while A ∂ , G − + n − 4.7 K T , n ∂Z ). Now, using the explicit form of the 2 A A , Z  1 + − ∂ ∂ nF B − 2 n Z ...Z n BC   − F 4.12 = 2 + 1 n 2 T H B B − 1 so one can apply to A n  F Z be an element of maximal degree ∂ ∂ variables, for the last term in ( λ + Z B − ∂Z ∂Z F, 3 b n + ,NF ∂ n D ABC A NF λ n B B  n n F 1) basis elements ∂Z – 30 – ABC b Z D ...A ,  Z Z 16 T )-tensor, cf. ( A 1 ∈ S = = nT to A R (2 Z 16 − = A n , , i.e. n F = 2 + 2 + n A so T n n α − = G G (3 ) = 2 T n   F = Y T gl Y n = ( − − AB ), etc. Eventually, for any element = Let F F n L A A ) we obtain G G Z Z C.1 ) can be equivalently rewritten in terms of the singlet vari- . βB   of degree 1 2 1 2 ,NT 4.14 B.2 2 ∂Y in ( ( 4.1 ∂ = λ n αA = 0 ) = b A is of maximal degree D G n Z are operators introduced in ( T 1 ( ∂Y T − ∗ ) and ( F N  n ) αβ ) irreducible components, we find that ∗  G totally-symmetric Z R B.1 ( A , n . It can be decomposed as homogeneous F T (3 , and A 2 gl is Z Z n ⊂ , F  1) Introducing the Hodge dualized As an example of passing from 1. The expansion coefficients in the homogeneous part, , (2 − Here, the term that was applied to degree we get the trace decomposition Here, equivalence operator form a rank- so where where n C Canonical basis Proof of Proposition variables the relations ( ables as we have or, for the second term, JHEP04(2020)206 = ). 1)- ,... , ) 3 F 2 ), we (C.7) (C.8) (C.9) ,... , (2 λ 2 1 and b D , ,M )-tensor so , C.7 AdS ) can be 1 ) depicted (7.11) (7.10) (7.12) R A . One-row (2 , . ↵ , 1)-vectors,
o Y , , = 0 cients of 3.2 4 ··· ··· ··· (3 ,... ··· 1) ) 1) tensor. = 0 , 2 (2 , = gl m m , ` ( n (2 of the counting 1 † (2 so ) as , ) traceless totally- o so n A . nT ↵ n 4.7 Y = 0 U = ) while all subsequent n ) is isomorphic to the = 0 ) ), we conclude that the correspond to scalar 4 . m n =2,wherethe )-tensors ( . ( n of the universal envelop- 4.9 • 0 R ` )tensors.Ontheother (2) singlets (7.7) can be , 3 ...A M 4.13 3 ⌘ S `F sp 1) tensor with indices de- cients of (7.3) are (3 A ,M , m 2 gl = ) . Recalling now that the trace A (2 (2 ,NT encoding the information about ` 1 ...B o 1)-tensors ( m o =1andstudyquotienthigher 2 ( ]. The basis elements are organized , A from ( (cf. figure ) is defined as ( B λ ), cf the plot on figure T ) we can arrive at the star-product ) so that expansion coe [ 4.1 m (2 6 M 1 = 0 2 (  ) can be depicted as on the plot in n , can be represented in the while the extra index +2 ) encodes a rank- hs ) A so ,B T +2 ) 1 C.6 ) l M 4.15 1)) m m
( A m ( C.7 spanned by M n , S m 1), and each factor can be considered (2) and C.8 ( T ) and the rank decomposition ( , n 3 T 1) • ] = 2 ...A ,NF A 1 , sp (2 T S ` (2  A o =1 ( 1 as o(2,1) module is decomposed in a direct o F C.3 (2 l ,N ( M F 3 0 30  , η so U S > Young diagrams, dots totally-symmetric =0 ` X n 1 , 1) – 31 – ` M m k A U , ) = here encodes a totally-symmetric 2) act on = m , and the bar stands for the complex conjugation. (2 ( : n o F 3 erent ideals, including the maximal one. We show ` C T ...Z ↵ S M, A 1) traceful two-row rectangular tensor (7.10) can be m . ⇡ ( 2 1 , m n o the linear space of the quotient algebra ( =1. ) A a ...B (2 2 . 2) 1 ). Each term in ( Z o : , M Z elements, each tensor 6 ...Z n gives the rank of ,B First, let us rewrite the decomposition ( 3 4.1 ( resulted from decomposing a traceful two-row rectangle of (2 This is because the linear space 1 m 0 • n C.7 o ) k A 1) irrep given by a totally symmetric traceless ...A traceless totally-symmetric > . of the complex algebra , 1 ) defined by the equivalence relation ( T ...A m ∈ S Z ,... (2) and n ( 1 X A , where ` denote here rank- † (2 stands for multiplicity, cf. (7.2). Elements of linear space (7.11) in ( ,... 2 A † m,n is a finite-dimensional algebra. Therefore, in order to produce an T F 3.2 sp , 2 ` l 2]) yields totally symmetric massless (Fronsdal) fields of increasing is isomorphic to the symmetric tensor product of F 5.1 1 ) ∀ , . 1 ...A m = = I 1 , 1
mo Y / , 1 )-covariant basis in the universal enveloping algebra ) M, A ( +2 F ). 1) are depicted as length- R n . Now, recalling that the gauge equivalence operator reads )-module. The decomposition ( F m F . Irreps , , 2]) ( = 0 3 2:[ n F 0 =1case,any k R | , = 0 ,..., T , a (2 T (3 T • n (2) invariance condition (7.7) says that these tensors are of particular index 1)-tensor through 2 = ` 4.17 (1 , , gl (3 M so =¯ : ∈ S sp 2:[1 ` | hu and successively applying the relation ( (2 † gl 3 F =1 U (1 F ) . Thus, we arrive at the decomposition ( so F s • Y hc ∗ ∈ S N )intheauxiliaryvariables(7.3)areboth of traceless elements is graded with respect to the eigenvalues . The . The canonical basis in the higher-spin algebra ( denote a spin- ) It follows that a linear space of the algebra In the In what follows, we explicitly consider the case of of rank . Y λ F ( ` m H -th trace of aF µ ∀ 5 sum where a superscript can be depicted on the following plot: decomposed into one-row tensorsscribed because by two-row any Young traceless diagramidentically, while with those more with than a oneCivita single cell tensor, cell in see in the (2.2). the second second row row vanishes are dualized usingrepresented the as Levi- an infiniteT collection of one-rowThen, traceless one Young can show diagrams. that a Indeed, linear let space of components Here, irreps two infinite familieshigher spin of algebras. idealsglobal Our that symmetry analysis yield algebra also applies bothby to analogy finite- the with case the and case of infinite-dimensional of quotient 7.2 Howe dual realizationHowe dual of algebras F hand, the symmetry type. It followstraceful that tensors the with index resulting symmetry expansion described coe by rectangular two-row Young diagrams where the involution Gauging spins spin algebras correspondingthat to di infinite-dimensional higher spin algebra one should use non-maximal ideals. We identify Following the Proposition Combining the first-trace decomposition ( Thus, the quotient ( m − ...A 1 2 A C Thus, the universalcovariant enveloping basis shown algebra on figure visualized as the first horizontal linelines on form the plot the in ideal. figure where the index labels the tower of suchthe rank- symmetric get the general trace decomposition, Each term F ing algebra hence it is a figure operator Proof of Proposition decomposition ( space of traceless elementsand as counting stated operators in form an the algebra,space Proposition i.e. [ Figure 5 Young diagrams of length with some ( Figure 4 into an infinite series ofas rank- one-row Young diagrams of length JHEP04(2020)206 A T into ) 3 . Let ), the (D.1) n A (C.10) (C.11) (C.12) n and ,M D Z AdS + (7.11) (7.10) (7.12) A . The ba- (2 ↵ 1). In the
o Y m , cients of ··· ··· 1) ··· ··· 1) tensor. ), is spanned , (2 , = ) on traceless , = 2 (2 so † (2 totally-symmetric C.8 ) o ` 1)-generators so . A , ` ↵ C.11 A + 1) 1)-tensors. To this Y U (2 , 1)-modules, spanned V . N ) , so (2 ( correspond to scalar , . V (2 =2,wherethe so N · A • 0 ) of the finite-dimensional )tensors.Ontheother so (2) singlets (7.7) can be ) of ] = 0 V M Z + ⌘ ) ( sp 1) tensor with indices de- cients of (7.3) are ,M ,  ,... 5.2 m V . ,N − 2 2 · ) (2 (2 A 1)-covariant manner. , )-tensors. Z ...B o A o m =1andstudyquotienthigher , 1 2 Z ( T R n is defined as ( [ , − Z B , (2 ) so that expansion coe T M 1 into a single framework that allows (3 = = ( = , so +2 ), indeed carries an adjoint represen- ,B = 0 gl +2 ) A ∗ l M A A 1)) k ⊥ ] m ( + 1) m spanned by , M ∗ , S V 1), and each factor can be considered (2) and m Z Z 5.3 ] T , 3 ( n • ...A A · . Then, calculating ( ( 1 sp (2 S )
   , (2 A n o =1 ( 1 as o(2,1) module is decomposed in a direct o m A l ( 1) M ( ] = 0 n F 3 , 2 = T 30 U 1)-tensors form a single rank- [ D S Young diagrams, dots 1)-tensors, and are thus isomorphic to ) , =0 , (2 1 . Note that Young diagrams in each column indeed are finite-dimensional , ,Z – 32 – 1) m M m A k (2 ( ) , so n 5 A we can decompose the original variable (2 T [ 2) act on so m = T , and the bar stands for the complex conjugation. (2 T ( n : 1 2 so [ o B 3 U erent ideals, including the maximal one. We show totally-symmetric traceless C where Denoting the inner product between two arbitrary D 1)] ↵ S M, P 1) traceful two-row rectangular tensor (7.10) can be m , ≡ ⇡ n ( 2 1)-vector and commutes with the counting and trace ∈ , A o , =1. (2 a A ...B (2 Q 2) 1 , (2 F o , T so , M ∀ [ ,B . Second, the adjoint action ( resulted from decomposing a traceful two-row rectangle of A so AB (2 ) 2 A m ⊥ , we first elaborate the oscillator technique combining the • T o η C k , expressed in terms of the trace expansion ( m 1) irrep given by a totally symmetric traceless Z ) ( 1 2 1)-tensors in a manifestly of the complex algebra n , ] = 0 T ...A ), it reads , (2) and 1 m = T , where +  † (2 ( stands for multiplicity, cf. (7.2). Elements of linear space (7.11) 5.2 n A † is a finite-dimensional algebra. Therefore, in order to produce an , (1 sp l 2]) yields totally symmetric massless (Fronsdal) fields of increasing 5.1 F A D k 1 P ) A . so I · Z and the compensator vector mo 1)] = . When combined into vertical columns (of fixed sum Y T -th position on figure / 1 M, [ ( , ` 1)-covariant basis of the universal enveloping algebra m are depicted as length- A = Q , ) we find the standard Casimir eigenvalue F . Irreps (2 2]) 2:[ 0 =1case,any Z k | , (2 A ,..., a T T • so (2) invariance condition (7.7) says that these tensors are of particular index 2 C.9 [ (1 Z so ): , ( 1) spanned by rank- 2 M =¯ sp 2:[1 , C | ) hu † (2 m =1 (1 , which appears in the decomposition ( 4.12 ( ) n traceless totally-symmetric s n so • T Y hc . The )intheauxiliaryvariables(7.3)areboth n ( denote a spin- D It follows that a linear space of the algebra In the In what follows, we explicitly consider the case of Y ( m )-tensor on the aF 1) vectors as Civita tensor, see (2.2). represented as an infiniteT collection of one-rowThen, traceless one Young can show diagrams. thatsum a Indeed, linear let space of where a superscript can be depicted on the following plot: decomposed into one-row tensorsscribed because by two-row any Young traceless diagramidentically, while with those more with than a one single cell cell in in the the second second row row vanishes are dualized using the Levi- components Here, irreps two infinite familieshigher spin of algebras. idealsglobal Our that symmetry analysis yield algebra also applies bothby to analogy finite- the with case the and case of infinite-dimensional of quotient 7.2 Howe dual realizationHowe dual of algebras F hand, the symmetry type. It followstraceful that tensors the with index resulting symmetry expansion described coe by rectangular two-row Young diagrams where the involution Gauging spins spin algebras correspondingthat to di infinite-dimensional higher spin algebra one should use non-maximal ideals. We identify R Parallel/Transverse variables. Now, let us demonstrate that each copy , , (2 (3 so two parts, parallel and transversal to the compensator vector, D Twisted basis To show the propositionoriginal variables working with Lorentz where the central dotelements means by rank- us confirm this byadjoint computing representation the ( eigenvalues of the Casimir operator of transforms each index asoperators ( an This already shows the vector spaces module tation of end, we first noteby that the traceless elements to infinity) for fixed corresponding totally-symmetric traceless gl differ by two boxes, which corresponds to taking trace of Figure 6 sis elements are organized into horizontal lines of traceless elements (of ranks running from zero JHEP04(2020)206 - ). A q C V (D.6) (D.7) (D.8) (D.9) (D.2) (D.3) (D.4) (D.5) (D.10) totally- ...V n 1 . C -transverse q . . V A . C k n V n A = 0 nF . ...Z A B ⊥ k = 1 ,...  C 2 ...V k n ∂Z ∂Z , 1) can be represented 1 F Z ∂ totally-symmetric and 1 , ∂Z A p ⊥ , n · B ⊥ V , n = 0 , V , 0 = 0 i q 0  Z a q B ...A Z ...Z ( 1 A ) can be represented as A V 1 ,N p . A n V n V A B B ⊥ ⊥ F A k V D.9 = Z ...a = 1). = 0 1 = 0 q a = as A A k k ) = k n − ...Z F ∂ ...Z Z ...C F Z k n A B 1 V k k 1 1 ∂Z ( · ⊥ B ⊥ N C which is a rank- n A | k ( ⊥ ∂Z ∂Z  Z p k F ≡ n Z p N n || A ...B ...a we can define the directional derivatives 1 1 ...B ) = a 1 B ...A q 1)-tensor 1) one can see that the original rank- , : -longitudinal parts. The resulting coefficients B 1 ( F B || , – 33 – F , ,Z V A A f B A 0 n so that n  (1 A , F n V 2 , ∂ || = A ⊥ q = V so where Z = 0 q X B + = Π X p + ) = − ∂ = (0 V p k goes to A B ...Z ⊥ ⊥ · V ∂Z B A ≡ Z A 1 · δ n ( B A , ⊥ n A ⊥ ∂Z ∂Z V ) = n Z n Z A 0 = Z ...A F Z -transversality condition at any 1 Z ), the expansion coefficients can be taken to be ( n ≡ = Π B A A ) A n 0 F F A V V ...Z ⊥ ...A , D.2 Z ( 1 1 ∂ || A n A A ∂Z ∂ introduced above, the expansion ( F -transversal and F Any irreducible -transversal tensors with respect to the first group of indices and Z 0 A + n A ≡ Z V V ) = ⊥ A ⊥ ) = A )-tensor ...A k ∂ ∂ 1 ⊥ R in Z A , Z ( satisfy the 1)-tensor. On the other hand, any element depending only on the parallel are = ( F , n n -longitudinal piece would lead to a vanishing contribution. Choosing the com- (3 q p n A F A (1 F gl 1)-covariant way as ∂ ...A ...C , ...B V 1 ) = ) 1)-scalar. Indeed, introducing the notation so 1 1 q , A ( (2 C Z B | ( F f p (1 so n F Now, we can decompose any element of degree Tensor analysis. so ...B 1 B Using the variable where Going from thecients first equality toF the secondlongitudinal one, in the we second group decompose of indices the (i.e. the expansion tensors are coeffi- proportional to we find that variables is a By construction, due to ( since any pensator in the standardsymmetric form traceless Another relation useful in practice is in an They satisfy the following relations Introducing the projector Π As a consequence, one obtains the transversality relations JHEP04(2020)206 F ∼ . In k (D.14) (D.15) (D.11) (D.12) (D.13) k A  0. The F , thereby 1)-tensors , , , expanding AB , which are 1 ) 0 (1 and F yields F is the projec- F k > B . k a 2 so fZ Z F F ∼  traceless totally- AB 1)-components: a + = 0, i.e. the scalar 2 + ⊥ , ∼ ⊥ k . For Z A f ⊥ B ⊥ 2 F (1 1 2 ⊥ Z A Z F + 1)-tensor so A F , − )( f A 2 k 0 AB (2 and ≡ f Z Z , and so 1 a  A k + AB F F Z ⊥ η A ⊥ k ). A 1)-trace operator decomposes as F ) is a sum over Fourier modes ∼ Z , , F parts we get + n )( (2 ⊥ k + A d 0 ⊥ D.11 f , = 0, we see that the expansion coeffi- so F D.11 -transversal, and Π A Z ⊥ F ⊕ n A , AB Z  B and F ⊥ V V ab A Π ⊥ A ... are of dimension two for  , A F 2 1 ⊥ F Z 2 0 V ⊕ k ⊥ A − 1 is of dimension one (and carries the trivial d in ∼ F + d fZ ) = )  0 A – 34 – AB A k ⊥ . This is the obvious analogue of the standard d ⊕ B + ◦ ⊥ Z AB ) = 0 we find that f 1)-module spanned by rank- Z F 0 0 , F Z + 2 = , n d A Z ( + ( 1 ) goes to (1 ) 2  V = A A ⊥ ⊥ F AB so − B ⊥ 1) branching rule ( n Z f Z -traceless and +  , Z D.13 A D 1)-modules )( 1) branching rule ( A ⊥ (2 f , k AB , A ⊥ AB Z η F (1 so F (2 , . . . n 1 is so + so ⊂ , ⊥ AB = ( thereby reproducing the branching rule ) + 2( 0 ◦ ⊥ A F , 1) B ⊥ ⊂ and the variable B 1 F AB ,  Z ◦ f Z − A (1 A are irreducible over the real numbers but become reducible over ⊥ A 1) = ( F 1) which corresponds to a traceless rank-2 Z ) = , so Z k Imposing the trace condition , . . . , n , A Z d 2 (1 ( ,..., AB , (2 Z AB 1)-tensors of ranks 1 and 0. The analogous procedure for f 1 2 so A , F ( , so F . Disentangling the resulting system, one finds all irreducible F + 1 (1 k -transversality condition we can decompose ≡ ) become related to each other since the n A denotes an irreducible  = 0 ) = so (3) branching rule where spherical harmonics are decomposed in terms of eigen- ) = − V ). Fixing the compensator in the standard form we get Z k + k ). Then, the expansion ( so ( Z D.10 d 2 ( n, ⊥ )-tensors of respective ranks 0, 1, 2. 1 ⊂ F 1)-modules − R  D.3 F Imposing the trace condition By way of example, let us consider lower-rank elements D.10 , , = = (2) (1 (2 In other words,rank-2 the traceless solution tensor, to aresentation the vector, of and trace a constraint scalar. isdemonstrating All the given together by they form an irreducible rep- where the firsttor term ( component arising as the tracethe is rank-2 not tensor independent. canwith By be the appropriate made linear change traceless. of basis, Using the trace relation obtained above along with the expansiongl coefficients being cf. ( which are k so vectors of the angular momentum along the vertical axis. both the coefficient where symmetric Lorentz tensors.representation) while Note that the so the complex numbersthis where way, they the split into the Fourier mode of eigenvalues cients (  of ranks Branching rules. JHEP04(2020)206 k as N ,... and 0 2 Z , (D.20) (D.21) (D.22) (D.16) (D.17) (D.18) (D.19) ⊥ . Such can be 1 1 , 0 F , encoded Z 0 = 0 ,... ] = 0, we find n 2 0 , 1 ,Z ,  ⊥ . Such a represen- is the Pochhammer is the eigenvalue of 2 ⊥  ⊥ 2 = 0 Z ] k Z m k ; ) 56 1 2 , x + k , degree which are however all  k , 1] = 0 | ), where 1 ) n ⊥ ⊥ (where ( − F ⊥ 0 , Z kf Z ⊥ 2 ( ⊥ ( n 0 . k  k Z = | f Z = 4 , n ) k k . | f m ⊥ n ) 0 N ) and noticing that [
f Z ) = 0 2 ⊥ ∂ 2 ( 1)-tensors, and ⊥ ⊥ Z ) in powers of the parallel variable variables while the expansion in ∂Z , Z n Z ( ( . 0 but a power series in f variables are naturally separated. Moreover, Z D.17 k (1 ( F 0 ⊥ Z g 0 k :[ > so F m 0 1) Z – 35 – n X = ,...
> k 2 1 2 k X − ,N k , ) with no definite N and +1 1 N ⊥ ) = + ⊥ m , ⊥  Z Z ) = = 0 k − ( ( ( singled out by the modified trace constraint, 4 ⊥ k k ! | = 0 F 3 f Z n 1)-tensors with ranks running over e ( S m ) k f , Let us introduce two commuting operators n 2 ⊥ ⊥ f 1 and (1 =0 Z ∞  . . ( X e m so − ) with expansion coefficients given by [ k H ⊂ k with ). Each such collection is labeled by the eigenvalue g ⊥ N = ⊥ k 5.2  Z
is a polynomial in ( D.18 2 ⊥ k | Z F n f ) are graded with respect to the eigenvalues of the counting operator k g Z ( traceless totally-symmetric F ) are arbitrary power series in k ) is convenient because ⊥ Z ( ), n D.18 f In conclusion, the solution space can be organized as an infinite number of collections of Imposing the modified trace constraint ( to produce elements D.10 variables. To this end, let us expand ⊥ bounded from below by traceless totally-symmetric in the polynomials of the counting operator denote the infinite seriessymbol) of which trace can creation be operators collective packed trace into the operators confluent actN hypergeometric on limit traceless function elements encode rank- where the irreducibility conditions the solutions which is now given by solutions of the form ( where assumed finite, i.e. tation ( This is essentially the two-dimensional Klein-Gordon equationknown. whose solutions are explicitly Ink what follows wein describe ( the solution space using the technique of and consider the subspace Proof of Proposition JHEP04(2020)206 1)- ) in , , the (E.1) (2 , (D.24) (D.25) (D.23) so D.11 4.1 5.2 N .... ] is given by ⊕ λ are shown in green [ ) ) which are packed n 3 • • • • • • • • • • hs ⊥

d

D • • • • • • • • • •

Z

• • • • • • • . ( • • ⊕ • k 2 | 8 n d 1) (standard vs modified) f , ⊕ (2 1 d so 1)-modules , ⊕ can be decomposed as ( • •

0

(2 , • • n d ). Thus, we obtain a different de-

• ( so • B ⊥ D . 1)-tensors , ∂ ⊕ 2 , , A ⊥ 1)-covariant but with respect to distinct ) (1 . D.21 k , n 1)-covariant basis of 2 Z d , 7 so d . The (2 N W C )–( k has a fixed eigenvalue on all elements inside a (1 ] as formulated in the Proposition d =0 ⊕ + ∞ so V λ k ] is given in the canonical basis by M k =0 [ 1 ∞ so ] in the canonical form. Each Young diagram of n λ M d N N λ [ D.20 ∼ = [ BBB hs – 36 – BIG ABC = ⊕ + hs n hs  . Since each , an 0 e ⊥ k − H W d 4 d ( N 1)-module = ), cf. ( basis shown on the plot figure , ⊕ = ⊥ L ) (1 Z 1 Baas . 0am ( N d so n f ⊕ Let us now recall that, according to the Proposition is the solution to the modified trace and counting con- twisted subspaces, see figure 0 1) representations d , n n 1)-modules ( BBB BBB , D W (2 ⊕ (1 1)-components before imposing any ) comprises all the traceless , 1)-covariant basis of 0 so 1)-covariant basis of so ①BM , BBB n d , (1 (1 (1 W so 0am BBB = ( so so - n ). This is the shown on the plot figure D n =0 ∞ D.16 D . The n M stands for an irreducible = k The canonical and twisted bases are each However, the Lorentz modules can be reorganized by introducing an infinite number H E Twisted-like We assume that the Lorentz generator is uniquely realized by the homogeneous operator representations. Note that they areinfinite sums. related by This a triangular transformation triangular wouldspace linear involve finite of transformation sums involving Lorentz if it weretrace done condition, in the as is manifest for instance in figure where each subspace straints ( where each space inside a generating function scription of the Terms in brackets form of subspaces Canonical vs twisted basis. linear space of themodules higher-spin algebra terms of irreducible Figure 7 length contours. The countinggiven operator contour. JHEP04(2020)206 N ). ∈ ) the (E.3) (E.4) (E.5) (E.2) n h ) (E.6) 5.24 − ⊥ ⊥ ,N  k , where N n, h ( N g − − − ⊥ k .  N • • • • • • • • • •

• • • • +1) • • • • • •

• = ( • • • • = 0 • • • • • ) of the counting opera- ⊥ ⊥ G F ,N ) k ,N h k N − ( N g ( ⊥ f • • 

• • ) = = are shown in red contours. The counting

.

• • A , ⊥ ⊥ n , G f A ⊥ W A ⊥ ,N f ∂ ∼ Z k 0 ] 1) algebra commutation relations ( + g g ∂ N is diagonalized on the subspace singled out , G N and ( ( A , ⊥ ) and ˙ g + (2 A A ⊥ ) and the operators ] = ] = ˙ P T so ] in the twisted form. Each Young diagram of length N g Z – 37 – A [ ). BBB , BIG A , g , g λ ,N , P [ ) P k A A = ⊥ ⊥ , . Subspaces ⊥ E.3 hs ∂ L N Z and the modified trace operator [ k A + [ ( = 0 d P 2 g n ,N k = ( L F − ) = N k A Baas . 0am ( = n g satisfy the T N g preserve two irreducibility conditions: they weakly commute 2 − A − A C k P P respectively. More explicitly, 1)-module 1) N , BBB ( BBB ⊥ − ) is some counting operator (e.g. a constant), i.e. (1 and Z and ⊥ ⊥ so L L ①BM BBB ,N ,N 1)-covariant basis of k k and , 0am N BBB N (1 ⊥ ( ( - ∂ g h so = has a fixed eigenvalue on all elements inside a given contour. irreducible ) = h k k ). The coefficient functions are fixed by the requirement that ⊥ . The N ,N by the irreducibility conditions ( weak commutativity property then reads with the weight operator and Denoting the generators the generators k 5.20 The generators The Casimir operator N • ( 0 is a rank- (B) g (A) where by dotted andfunctions primed with functions we denote the results of commuting the counting Let us introduce the notations It turns out that, undertwo the properties: above condition, the resulting generators possess the following are defined by sometors functions ( Figure 8 k operator while the transvections JHEP04(2020)206 i k g N ] = 0 (E.7) (E.8) (E.9) h (E.10) (E.11) − . ⊥ h  ⊥ − , 2 ⊥  L ⊥ N A ⊥  ∂ ) , ¨ f E 0, while the counting + 23 + 2) + ˙ + f ≈ 1) algebra on any elements ⊥  yields an equation on , ) ) (2 0 N ) of the Casimir operator , we obtain [ 0 enter the generators only (2 ⊥ h )( − gg ). Such a constraint singles out 0 (the weak equality symbol so gg ˙ A || g N − E.9 Z ⊥ f Z + ( ≈ + ) depend only on the transverse ⊥  2 k F 2 + n  g A ⊥ g is not diagonal after imposing the N ( ..., ..., ( ( Z − 2 j fg . ) − + + k g − C g 2 2 ⊥ ⊥ N ⊥ ⊥ + ( = + ¨ = 1  N N  1) algebra are closed. On the other hand, this j ⊥ g 2 2 ˙ , g g ) hold) is satisfied automatically since   f g N f (2 ) 2( ˙ ) ˙ 0 f + + so − . − E.3 f 0 ⊥ gf ⊥ A ⊥ ) and A ⊥ – 38 – + are satisfied provided that k N (otherwise, N ∂ + gf 1 1 2 ˙ ) h ∂ N f g ⊥ f ( g f i gf ( N + + f + ¨ − − 0 0 g = g f A ⊥ 2 ⊥ ) + ( i 0 + ˙ Z Z f = = 0 ) which are required to span the  g gg g -dependent functions, while f − ) in each order in the variable gh A 2( ⊥ E.2 + Z = − 2 − E.7 g 2 the right-hand side can be made proportional to for the counting constraint C )( yields ) should weakly vanish. The form ( ] = ˙ ). Otherwise, the first condition imposes a constraint claiming that a particular f h . f k f, g E.9 − 5.20 + N commutation relations itself. For the modified trace constraint ⊥ 2 k  f 1) , , N A must be independent of (2 = ( P . [ E
E property (A) property (B) so 1) means equal when the constraints ( , . In other words, there is a single algebraic relation for two functions. A particular (2 trace and weight constraints). purely depends on thehand-side counting of operators. ( suggests The that first the group traceterm of constraint must terms be on the right- where For particular The The ≈ commutes with any through and The k so f Now, the condition ( Assuming that the coefficient functions are power series in the counting operators we In particular, this condition demonstrates why the coefficient functions are chosen to be independent of • • • U 23 the trace operators in ( combination of tracea operators subspace must where act theis trivially commutation distinct on relations from any ofof the element the generators ( where the expansion coefficients counting operator and can represent them in the form Now, we consider the three conditions formulated above in turn. JHEP04(2020)206 ) 1) , E.7 (2 ) + 1. (E.16) (E.17) (E.18) (E.19) (E.12) (E.13) (E.14) (E.15) n so ). Also, ( 0 ). f is given by E.6 6.1 N can be set to ) = in ( ∈ n ζ 0 ( , n 0 ∆ ≈ = 0, etc. Hence, ( ) 2 g ζ ) drastically simplifies to − E.8 = 0, ˙ ⊥ . 0 . The other half parameterises  C , the constant g = 1. Recalling that the j . ( 0 g
∈ A ζ ⊥ ..., ∂ ζf ) suggest that the trace condition ζ parameterized by + ∀ 2 ζ . , . n and 2 f , E.9 0 ) + 1
i ) should give the expected eigenvalue. n = f ) n , ≈ 1) generators are given in ( f 2 , i.e. ( n , 0 k ⊥ α 0 g + ( + 1 0 (2 N  = 0, etc. E.10 0 ≈ ⊥ + f ζf ) of the same form as in the continuous-spin field ). Using the trace constraint we see that the 1) and introducing for the conformal weight ( 0 A where ⊥ so 2 N − n ζf f ) ζ 1 − E ˙ 1 , f∂ E.9 n E.14 − α 0 1 ( 2 − – 39 – ζ 0 ) = ζ , − + ζf ). ⊥ − − ζ n ζf ( 0 ).  − A Then, the commutator ( ⊥ = = α ) = E.7 ∆ ≈ E ⊥ = f g∂ . The dual (shadow) weight is then 25 n ˙ 2 ] 1) generators since such a freedom can be absorbed by E.14 N f ) = 2 , ,N that can be solved as − n k C ∈ ( (2 [ ) we get 1 i 0 N o − ( f ] = , ζ i g 2 is a constant function ζ α E.13 = . Therefore, from now on we set 0 − 24 0 f ) and the Casimir operator ( f ⊥  ). and the counting constraint , ) and ( E.12 A 2 C ) is arbitrary, as follows from the definition of the prime ) we find out that it can be identified as ζ P k n [ ( E.12 N ≈ ( ∆ 0 ⊥ f is an arbitrary constant. ) we directly find that  ), it follows that all derivative functions are zero: = 2 ζ 0 E.6 f E.13 = Finally, consider the Casimir operator ( The solution ( A convenient choice h The other (opposite) choiceIn would be this a way, constant we function arrive at the modified trace ( 24 25 with arbitrary coefficients The resulting expressions for the twisted-like theory (see appendix Casimir eigenvalue is parameterized asthe ∆(∆ notation where constraint Since the only invariantunity combination or absorbed of into parameters is first term vanishes. Then,Indeed, the using second ( term All in all, we find that the Casimir eigenvalue on each subspace singled out by the trace yield the required relation which means that the transvectionsby have the a modified well-defined action trace on constraint the ( subspace singled out from ( should be given as i.e. From ( reduces to the relation where equivalent realisations of the linear changing basis. Thus,then one fixed of by functions means of can the be relation chosen ( arbitrarily while the other is solution fixes one half of the expansion coefficients JHEP04(2020)206 ] , A 11 , , Int. J. (2016) D 94 , ] 07 JHEP (2014) 122 J. Phys. , ]. ]. , Phys. Lett. 10 , JHEP ]. Phys. Rev. , SPIRE SPIRE , , April 7, 2015, , May 27, 2015, IN arXiv:1711.08482 IN ]. [ JHEP [ [ , SPIRE KITP KITP IN arXiv:1311.7413 ][ SPIRE [ IN ][ (1985) 343 , talk at , talk at . . 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