Bosonic Fradkin-Tseytlin equations unfolded

We test infinite-dimensional extension of algebra suk,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{s}\mathfrak{u}\left(k,\;k\right) $$\end{document} proposed by Fradkin and Linetsky as the candidate for conformal higher spin algebra. Adjoint and twistedadjoint representations of suk,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{s}\mathfrak{u}\left(k,\;k\right) $$\end{document} on the space of this algebra are carefully explored. For k = 2 corresponding unfolded system is analyzed and it is shown to encode Fradkin-Tseytlin equations for the set of all integer spins 1, 2, . . . with infinite multiplicity.


Introduction
In this paper we study unfolded formulation of Fradkin-Tseytlin equations [1] Π ∂ µ · · · ∂ µ s φ ν(s) = C ν(s),µ(s) , ∂ µ · · · ∂ µ s C ν(s),µ(s) = 0 , (1.1) which describe free conformal dynamics of spin s traceless field φ ν(s) in 4-dimensional Minkowski space. Here C ν(s),µ(s) is associated with traceless generalized Weyl tensor separately symmetric with respect to group of indices µ and ν and such that symmetrization with respect to any s + 1 indices vanishes, Π is a projector that carries out necessary symmetrizations and subtracts traces. Generalized Weyl tensor C ν(s),µ(s) is obviously invariant with respect to gauge transformations with traceless gauge parameter ν(s−1) and Minkowski metric η νν . If full nonlinear conformal higher spin theory exists, these equations should correspond to its free level. As nonlinear AdS higher spin theory teaches us, the main ingredient needed to construct such kind of theories is higher spin algebra that describes gauge symmetries JHEP12(2016)118 of the theory. In paper [2] Fradkin and Linetsky proposed a number of candidates for the role of infinite-dimensional 4d conformal higher spin gauge symmetry algebra, which extends ordinary 4d conformal algebra 1 so(4, 2) ∼ su (2,2). Their construction is based on the oscillator realization of su (2,2) [3,4]. Here we give a straightforward generalization of their results for the case of algebra su(k, k) with k ≥ 2 and briefly discuss the structure of the infinite-dimensional algebras obtained.

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Consider the involution † of the star product algebra defined by (1.10) Real form of gl(2k, C) singled out by the requirement is identified with algebra u(k, k). General element of u(k, k) has, thus, form X u(k,k) = X α β L α β +XαβLαβ + X αβ P αβ + X αβ K αβ + XD + X Z , (1.12) where X α β andXαβ are mutually complex conjugate k × k-matrices, X αβ and X αβ are Hermitian k × k-matrices and X, X are real numbers. Algebra u(k, k) decomposes into direct sum u(k, k) = su(k, k) ⊕ u(1) , (1.13) where u(1) is spanned by Z.
To construct an infinite-dimensional extension of u(k, k) let us bring all polynomials (not only bilinear) of oscillators (1.3) into the play still requiring them to be centralized by Z [Z, f ] * = i 2 (n a − nā − n b + nb)f = 0 (1.14) and to satisfy reality condition (1.11). Corresponding Lie algebra with respect to commutator (1.7) was called iu(k, k) in [2], where letter i means infinite. Decomposing general element of iu(k, k) into a sum of traceless components multiplied by powers of a · b andā ·b and taking into account that a · b = D − iZ,ā ·b = D + iZ one gets (1. 15) Here f u,v is traceless with respect to a , b andā ,b, i.e.
(1. 17) Here ideal I m is spanned by the elements of the form (1.18) where dots on the right-hand side denote the lower power terms.
(1. 19) Algebra isu 0 (k, k) is semi-simple, in what follows we omit index 0 and denote it as isu(k, k). Let us note that algebra iu(2, 2) is isomorphic to AdS 5 higher spin algebra that was discussed in several papers [5][6][7][8][9][10]. In [8] it was denoted as cu(1, 0|8) where 8 indicates the number of oscillators used and pair 1,0 points out that it has trivial structure in spin 1 Yang-Mills sector. Algebra isu (2,2) was originally (in [2]) denoted as hsc (4), where hsc means higher spin conformal and 4 indicates that it extends 4-dimensional conformal algebra. It is isomorphic to the minimal AdS 5 higher spin algebra denoted as hu 0 (1, 0|8) in [8]. As was discussed in [11][12][13] one can associate the minimal AdS 5 higher spin algebra with the quotient of universal enveloping AdS 5 Lie algebra over the kernel of its singleton representation.
In the present paper we analyze unfolded system corresponding to algebra iu(2, 2) and show that it describes a collection of Fradkin-Tseytlin equations that corresponds to all bosonic spins with infinite degeneracy. Let us note that some other approaches to Fradkin-Tseytlin equations were suggested in papers [14,15].
The rest of the paper is organized as follows. In section 2 we recall some relevant facts about unfolded formulation. Structure of algebra's su(k, k) adjoint representations on the vector space of iu(k, k) is discussed in section 3. In section 4 we study twistedadjoint representation of su(k, k). In section 5 unfolded formulation of conformal higher spin bosonic equations is analyzed for k = 2. Section 6 contains conclusions. In appendix A we recall relevant facts concerning finite-dimensional sl(k) ⊕ sl(k) irreps. In appendix B we find basises where adjoint and twisted-adjoint modules from sections 3 and 4 decompose into submodules. In appendix C σ − andσ − -cohomology corresponding to the gauge sector and Weyl sector of unfolded systems under consideration are found.

Unfolded formulation: preliminary remarks
Let M d be some d-dimensional manifold with coordinates x 1 , . . . , x d . Any dynamical system on M d can be reformulated in unfolded form of the first order differential equations [17] (see [18] for a review) dW Ω = F Ω (W ) . (2.1) Here W Ω (x) is a collection of differential forms (numerated by multiindex Ω) of ranks deg(W Ω ) = p Ω , d is exterior differential and is a p Ω + 1 rank form. F Ω (W ) is composed from elements of W Ω (x), which are multiplied by virtue of exterior product 2 and are contracted with constant functions In this paper all products of differential forms are supposed to be exterior and we omit the designation of exterior product ∧ in formulae.

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Compatibility conditions of (2.1) require F Ω (W ) to satisfy identities where the left-hand side of (2.5) is (anti)symmetrised according to (2.3). Any solution of (2.5) defines a free differential algebra (FDA) [19]. In what follows we assume that (2.5) holds independently of the value of space-time dimension d. In this case FDA defined by (2.5) is called universal. Unfolded system (2.1) corresponding to universal FDA is invariant with respect to gauge transformations where Ω (x) are p Ω − 1-form gauge parameters. Let us analyze system (2.1) perturbatively assuming that fields of zeroth order form a subclass of 1-forms W A (x) ⊆ W Ω (x). The most general form of F A (W ) in the sector of zero order fields is where constants f A BC = −f A CB due to (2.5) are required to satisfy ordinary Jacobi identities. Therefore W A (x) can be identified with connection 1-form taking values in some Lie algebra g with structure constants f A BC and system (2.1) reduces to the zero curvature condition Gauge transformations (2.6) become usual gauge transformation of a connection 1-form in this case where C (x) is 0-form gauge parameter. Let us treat all other fields from W Ω (x) as fluctuations of W A (x). For our purposes it is sufficient to consider the case when W Ω (x) consists of 1-forms ω a (x) and 0-forms C i (x) only (the general case is considered in [20]). System (2.1) linearized over W A (x) reduces to as a consequence of zero curvature condition (2.8).
As it was argued in [20] the term on the right-hand side of (2.10) should belong to nontrivial class of 2-nd Chevalley-Eilenberg cohomology taking values in g-module M ⊗M * . Indeed, compatibility of (2.10) in the sector of C i (x) is equivalent to the closedness of (2.13) with respect to Chevalley-Eilenberg differential (see [21]) are matrices acting on module M ⊗M * . If (2.13) is δ ChE -exact the right-hand side of (2.10) can be removed by the field redefinition And conversely if some field redefinition removing the right-hand side of (2.10) exists, it should necessarily have form (2.18) with W A θ A a i satisfying (2.17). System (2.8), (2.10) and (2.11) is locally invariant with respect to gauge transformation 3 (2.9) of connection 1-form W A (x) and the following gauge transformations of fields ω a (x) and There are also gauge transformations with parameters associated to ω a , which are discussed later (see (2.24)).

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of zero curvature condition (2.8) is fixed, the gauge symmetry above breaks down to the global symmetry that keeps W A 0 stable. Parameter of this symmetry A 0 (x) should obviously satisfy equation which is consistent due to zero curvature condition (2.8). Equation (2.21) reconstructs A 0 (x) in terms of its value A 0 (x 0 ) at any given point x 0 . So A 0 (x 0 ) plays a role of the moduli space of W A 0 global symmetry algebra, which therefore can be identified with g. When substituted to (2.10), (2.11), W A 0 plays a role of vacuum connection describing g-invariant background geometry. We only require component of W A 0 corresponding to the generator of generalized translation (i.e. generalized coframe) to be of maximal possible rank. Let us consider system (2.10), (2.11) with W A = W A 0 substituted.
As follows from above consideration it is globally g-invariant with respect to transformations (2.19) with W A = W A 0 and A = A 0 substituted. This system is also gauge invariant with respect to gauge transformations where a (x) is 0-form gauge parameter associated to field ω a (x).
To analyze dynamical content of system (2.22), (2.23) let us first suppose that the righthand side of (2.22) is zero. In this case equations (2.22), (2.23) are independent and both have form of covariant constancy conditions. Suppose that modules M andM are graded with grading bounded from below. Decompose covariant derivatives (2.22), (2.23) into the summands with definite gradings. We assume that each covariant derivative contains a single operator of negative grading (the case when there are several operators with negative gradings was considered in [22]) (2.25) Here D 0 ,D 0 denote operators of zero grading which include exterior differential, σ η + ,σ θ + denote purely algebraic operators of various positive gradings and σ − ,σ − are purely algebraic operators of negative grading. Operators σ − andσ − are nilpotent due to the nilpotency of covariant derivatives (2.12).
Let subspace of M with fixed grading n be called the n-th level of M. Analyzing equation (2.22) and its gauge symmetries (2.24) level by level starting from the lowest grading one can see [23] that those fields which are not σ − -closed (they are called auxiliary fields) are expressed by (2.22) as a derivatives of the lower level fields. Here space-time indices of derivatives are converted into algebraic indices by virtue of coframe. σ − -exact fields can be gauged to zero with the use of Stueckelberg part of gauge symmetry transformations (2.24). Leftover fields (that are called dynamical fields) belong to H 1 σ − the 1-st

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cohomology of σ − . We also get that differential gauge parameters (i.e. those that do not correspond to Stueckelberg gauge symmetry) belong to H 0 σ − . Let E n denote the left-hand side of (2.22) corresponding to the n-th level. Suppose equation E m = 0 is solved up to the n − 1-st level inclusive, which means that all auxiliary fields up to the n-th level properly expressed in terms of derivatives of dynamical fields. Bianchi identities D 0 D 0 ω ≡ 0 (2.26) on the n−1-st level require E n to be σ − -closed. If H 2 σ − the 2-nd cohomology of σ − is trivial on the n-th level, equation E n = 0 can be satisfied by appropriate choice of auxiliary field on the n + 1-st level. In other case E n = 0 also imposes some differential restriction on dynamical fields requiring that E n belongs to the trivial cohomology class. Therefore nontrivial differential equations on dynamical fields are in one-to-one correspondence with H 2 σ − . Moreover, if h ⊂ g is a subalgebra of g that acts horizontally (i.e. keeps levels invariant), differential equations imposed by (2.22) and H 2 σ − are isomorphic as h-modules. Summarizing, the dynamical content of equation (2.22) with the zero right-hand side is described by H 0 σ − , H 1 σ − , H 2 σ − which correspond to differential gauge parameters, dynamical fields and differential equations on the dynamical fields respectively. Analogously for equation (2.23) the dynamical fields and differential equations correspond toH 0 σ − andH 1 σ − . To analyze system (2.22), (2.23) with nonzero right-hand side let us consider operator (2.28) In these notation system (2.22), (2.23) can be rewritten in the following form where new fieldΨ is a pairΨ = (ω, C) incorporating 1-forms ω and 0-forms C. Here all operators are extended by zero on the spaces where they undefined. OperatorD 0 is nilpotent due to compatibility conditions of system (2.22), (2.23). Gauge transformations (2.24) take form δΨ =D 0Υ , (2.30) whereΥ = ( , 0). Let us considerσ − -cocomplexĈ = (Ŝ,σ − ) with p-form elementΨ p ∈Ŝ defined as a pairΨ p = (ω p , C p−1 ), where ω p is p-form taking values in module M and C p−1 is p−1-form taking values in moduleM (C −1 ≡ 0). Standard definition ofσ − -closed p-forms subspacê C p = (ω p C , C p−1 C ) gives in components the following relations (2.31)
In section 5 we use above technic to analyze dynamical content of su(2, 2)-invariant unfolded system that was originally introduced in [16]. Before it we explore structure of underlying su(k, k)-modules.

Structure of adjoint module
Consider adjoint action of algebra su(k, k) on the vector space of algebra iu(k, k), which is given by commutators (ad X u(k,k) ) = [X u(k,k) , ·] = 2X u(k,k) ← → ∆ . We have where n a , n b , nā , nb denote Euler operators counting the number of corresponding variables. Let M ∞ denote corresponding su(k, k)-module.

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commute with adjoint action of su(k, k) (3.1). Moreover due to centralization requirement (1.14) Modules M ∞ and M ∞ s are reducible with submodules where I m is ideal (1.18) and Note that I m s ≡ 0 for m ≥ s.

Consider quotient modules
where and where the number m + 1 on the right-hand side of (3.11) indicates multiplicity of modules M s . The basis where decomposition (3.9) becomes straightforward has form where subset with the fixed value of s corresponds to the basis of submodule M s ⊂ M ∞ s . Here g v s (Z, D) is homogeneous polynomial of degree v in two variables Z and D, which particular form is found in appendix B.1. It is important to note that in (3.12) Z and D are treated as a new variables independent on oscillators. Elements m s −v (a, b,ā,b) are traceless (see (1.16)) eigenvectors of operators s 1 and s 2 (3.2) corresponding to eigenvalue s − v

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with some reality conditions discussed later. In other words m s −v are monomials of the form ;β(nb) ;α(nā) are traceless complex tensors symmetric separately with respect to each group of indices α(n a ) ,α(nā) , β(n b ) ,β(nb), where number in parentheses indicates the number of indices in the group, and a α(na) = a α 1 · · · a αn a denotes n a -th power of oscillator a and analogous notation for oscillatorsā, b andb. Certainly values of n a , nā, n b , nb in (3.14) should be coordinated with s and v through formula (3.13).
Due to above arguments B v s,s forms, with respect to generators (ad In what follows we study the structure of module M s and in particular show that it is irreducible. Elements L α β ,Lαβ , D , Z of u(k, k) commute with Z s−s g v s and therefore are represented in M s by the same operators as in module M ∞ s . As shown in appendix B.1 elements P αβ and K αβ are represented in M s by the following operators (up to an overall factor i s Z s−s ) where n = n a + n b ,n = nā + nb, ϕ(n) = 1/(n + k) and Π ⊥ = (Π ⊥ ) 2 is projector to the traceless component (1.16) Every element B v s,s has a definite conformal weight  (3.19) and (3.20) with v = 0 have the lowest conformal weight −s + 1 and the highest conformal weight s − 1 correspondingly.
All the elements B v s,s that form basis of M s (i.e. with fixed s = 1, . . . s) can be arranged on the following diagram Here every dot (•) indicates some B v s,s . All dots in the same row correspond to B v s,s -s with the same conformal weight indicated on the left axis. Dots compose the collection of rhombuses, which are distinguished by the value of v = 0, 1, . . . , s − 1 indicated on the bottom of the diagram. The lowest (the highest) dot in each rhombus corresponds to B v s,s min (B v s,s max ) (see (3.19), (3.20)) of the lowest (the highest) conformal weight for given v. The arrows indicate transformations which change orders of B v s,s -s with respect to oscillators a, b,ā,b in such a way that s is kept constant and ∆ increases by 1. Namely : n a → n a + 1 , It is convenient to introduce independent 5 "coordinates" on diagram (3.21) Here v numerates the rhombus in (3.21) and q (t) indicates the number of upper-right (upper-left) arrows one should pass from the very bottom dot to get to the indicated dot.
indicate on the bottom, the right, the left, and the upper corners of rhombus v correspondingly. In these terms all other variables are expressed as

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Let us note that complex conjugation (1.11) transforms B v s,s (v, q, t) corresponding to the dot (v, q, t) toB v s,s (v, t, q) corresponding to the dot (v, t, q) symmetric with respect to reflection in a line connecting the top and the bottom of rhombus v. Therefore due to reality condition (1.11) coordinate-tensors (i.e. tensors like x in (3.14)) of B v s,s (v, q, t) and annihilated by (ad P ) αβ is B 0 s,s min given by formula (3.19) with v = 0. We, thus, conclude that module M s is irreducible.
From (3.16) one finds that quadratic Casimir operator

Structure of twist-adjoint module
Let us now consider twisted-adjoint su(k, k)-moduleM ∞ . It is spanned by oscillators a α , b α ,āα andbα, where oscillatorbα is obtained from oscillatorbα by twist transformation conserves. Due to conservation of commutator (4.2) operators that represent u(k, k) oñ M ∞ can be obtained from that of M ∞ by simple replacement (4.1). We have (cf.

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where nb is Euler operator for oscillatorb. We requireM ∞ to be annihilated by tw ∞ are operators that commute with (4.3). Note that due to (4.4)s 1 f =s 2 f for any element f ∈M ∞ . Twist transformation (4.1) applied to the basis elements of M ∞ s (3.12) gives rise to the following elements ofM ∞ is annihilated by twist-adjoint action of su(k, k) (4.3), and satisfy twisted tracelessness relations (cf.
In other wordsm s −v are monomials ; ;β(nb),α(nā) are complex traceless tensors separately symmetric with respect to upper and lower group of undotted indices and of the symmetry type described by tworow Young tableau with first(second) row of length nb(nā) with respect to dotted indices. Certainly values of n a , n b , nā , nb in (4.12) should be coordinated with s and v through formula (4.10).

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Due to above argumentsB v s,s forms, with respect to generators (tw In what follows we omit index 0 and denoteM 0 andM 0 s asM andM s , respectively. To find how P αβ and K αβ are represented in basis (4.7) one should apply twist transformation to (3.16), (3.17). We have Although all the above formulae were obtained by application of twist transformation (4.1) (which conserves commutators (4.2)) to the analogous formulae corresponding to the adjoint modules the structure of the twist-adjoint modules and their analysis have some important nuances in comparison with the adjoint case.

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Firstly, the twist-adjoint modules are infinite-dimensional. This is because operator s 1 contains the difference of n a and nb and, thus, requirement (4.10) does not bound the order ofm s −v with respect to a andb.
Secondly, contrary to the adjoint case elementsB v s,s of the twist-adjoint module are not linearly independent. Indeed, as was discussed abovem s −v forms with respect to the dotted indices sl(k)-module corresponding to the two-row Young tableaux with the first row of length nb and the second row of length nā (see (4.13) and appendix A). Therefore where (· · · ) are some coefficients.
In what follows let us reduce set (4.7) to linearly independent subset Let the elementsB v s,s and corresponding monomialsm s −v with v = v max be called terminal and let denote them asB t v s,s andm t s −v . Finally let us note that after we have reduced to linearly independent subset (4.22) it is possible to treatZ andD as independent of oscillators variables analogously to adjoint case.
For fixed s all linearly independentB v s,s can be arranged in the following diagram . . .  Here every dot (•) indicates someB v s,s . All dots in the same row correspond to different B v s,s with the same conformal weight (indicated on the left axis) which due to (4.19), (4.6) range for fixed value of v = 0, 1, . . . , s − 1 from Introduce independent coordinates on (4.23) where v numerates the stripe and q (t) indicates the number of upper-right (upper-left) arrows one should pass from the very bottom dot to get to the indicated dot. In these terms all other variables are expressed as with i sZ s−s g v s and therefore conserve set (4.23). Looking at operators (tw P ) αβ and (tw K ) αβ (4.16) one sees that their 1-st, 3-d and 4-s terms also conserve (4.23), but 2nd terms once act at the terminal elementB t v s,s decreases v max by 1, but keeps v the same and, thus, maps it toB vmax+1 s,s , which due to (4.21) is equivalent to the sum of terms corresponding to s − 1 , s − 2 , . . . So it looks like thatM s is not su(k, k)-invariant. In appendix B.2 it is shown that this is not the case, since these problem terms zero out.

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We, thus, shown that analogously to the adjoint case modulesM ∞ s ,M m s andM m admit decompositions (4.29) The basis ofM ∞ s coordinated with this decomposition has form where elements with fixed s , which are listed in diagram (4.23), form basis ofM s . In the same manner as for module M s one can show thatM s is irreducible with quadratic Casimir operator given by formula (3.27).

Unfolded formulation of Fradkin-Tseytlin equations
In this section we set the number of oscillators k = 2. According to procedure described in section 2 consider zero curvature equation ( and

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into Z s−s in front of M s basis (see (3.12)) and analogously for operatorσ ∞ andZ s−s . First consider σ ∞Zh , whereh is an arbitrary function of a α , b α ,āα,bα and ξ αβ . One has (5.10) Taking into account that the 3-rd term of (5.10) zero out if dotted indices in parentheses are antisymmetrized one has where From (5.11) one gets that for general power ofZ and analogous formulae for operatorσ ∞ . Therefore after field redefinition  Here ω s , C s ,C s are 1 and 0-forms taking values in corresponding irreducible modules and where in addition to Ξ αα andΞαα (see (5.8)) we define and tensors ε , ϕ , E, S are traceless. For cohomology ofσ − we have (up to an overall factor i s−s Z s−s ) where tensors C ,Ẽ ,S are traceless and symmetry type ofẼ (S) with respect to undotted indices corresponds to two row Young tableaux with the first row of length s − 1 and the second row of length 1 (2) (see appendix A for more details). Cohomology ofσ − are complex conjugate to (5.20). As one can easily see operator σ+σ mapsH 0 s ;2 +H 0 s ;2 to H 2 s ;0 . So to speak cohomology H 2 s ;0 is "glued up" by σ +σ. In other words 0-formH 0 s ;2 +H 0 s ;2 is not closed with respect to operatorσ and H 2 s ;0 isσ-exact. We, thus, have that 0-th and 1-st cohomology ofσ (see (2.31)-(2.33)) areĤ α(s +1) is the generalized Weyl tensor, which is expressed in terms of ϕ, finallyẼ α(s +1),γ ;β(s −1)δ = 0, Eα (s +1),γ ; β(s −1)δ = 0 are differential equations imposed on C andC. Direct form of these equations can be also easily obtained. We have Here symmetrization over the indices denoted by the same latter is implied and to avoid projectors to the traceless and/or Young symmetry components we rose and lowered indices by means of αβ , αβ , αβ , αβ . If transformed from spinor indices α,α to vector indices µ (by means of Pauli matrices) equations (5.22), (5.23) coincide with equations (1.1), (1.2) for spin s field. Here C α(2s ) andCα (2s ) correspond to selfdual and antiselfdual parts of C ν(s ),µ(s ) . We, thus, showed that system (5.16) realizes unfolded formulation of spin s Fradkin-Tseytlin equations.

Conclusion
We have proposed unfolded system (5.16) that describes linear conformal dynamics of spin s gauge field (spin s Fradkin-Tseytlin equations). We also have shown that any unfolded system based on su(2, 2) adjoint and twisted-adjoint modules M m andM m , M m , m = 0, 1, . . . , ∞ can be decomposed into independent subsystems of form (5.16) by means of appropriate field redefinition and found spectrum of spins for any m. In particular we have shown that system of equations proposed in [16] (5.6) describes conformal fields of all integer spins greater or equal than 1 with infinite multiplicity.
This work can be considered as a first modest step towards construction of the full nonlinear conformal theory of higher spins. One of the main ingredients of the higher spin theories is a higher spin algebra. Our results pretend to be a probe of different candidates to this role. We see that su(2, 2) modules M m s ,M m s ,M m s , m = 1, 2 . . . mediate between those with m = 0 and m = ∞. One can speculate that the same is true for algebras isu m (2, 2), which mediate between isu(2, 2) and iu (2,2).
Having in mind that conformal higher spin theory has to be somehow related to AdS higher spin theory one can suppose that algebra isu(2, 2) is more preferable since its spectrum just literally coincides with the spectrum of some AdS higher spin theory. On the other hand equations proposed in [16], which correspond to iu(2, 2), are considerably simpler than (5.16). Therefore an interesting question arises wether it is possible to simplify (5.16) maybe by mixing again gauge and Weyl sectors of the theory. Another important area of investigation is to consider super extensions of isu m (2, 2) and, thus, bring fermions into the play.

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Acknowledgments I am grateful to Mikhail Vasiliev for extremely useful discussions at all stages of the work. I would like to thank Vyacheslav Didenko for helpful comments on the manuscript. The work is supported in part by RFBR grant No 14-02-01172. u(k, k) is generated by L α β andLαβ (see (1.8)). Let the first summand of sl(k) ⊕ sl(k) that is generated by L α β be referred to as undotted sl(k) and the second summand that is generated byLαβ be referred to as dotted sl(k). All finitedimensional irreps of sl(k) ⊕ sl(k) are given by tensor products of finite-dimensional irreps of undotted and dotted sl(k). In what follows we recall some well-known facts about sl(k)irreps taking as an example undotted sl(k). Needless to say that the same arguments work for dotted sl(k) once undotted indices are replaced by dotted indices. Finite-dimensional irreps of sl(k) are given by sl(k)-tensors written in symmetric basis or equivalently by sl(k)-tensors

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where i-th row of the upper(lower) tableau corresponds to i-th upper(lower) group of totally symmetric indices of tensor T in symmetric basis (A.1) or equivalently i-th column of the upper(lower) tableau corresponds to i-th upper(lower) group of totally antisymmetric indices of tensor T in antisymmetric basis (A.2). Let Young tableau with rows of lengths λ 1 ≥ · · · ≥ λ k be denoted as Y(λ 1 , . . . , λ k ) and Young tableau with columns of heights µ 1 ≥ · · · ≥ µ h 7 be denoted as Y[µ 1 , . . . , µ h ].
Using totally antisymmetric tensors α[k] , α[k] one can rise and lower indices of T Taking into account that tensor product of two -tensors is equal to alternative sum of Kronecker deltas product where sum is taken over all permutations of (1, . . . , k) and π(σ) is the oddness of permutation σ, one can readily see that (A.6) vanishes if µ i +μ j > k. Indeed, in this case at least one of Kronecker deltas in every summand of (A.7) contracts with T that is considered to be traceless. Therefore, the irreducibility conditions above are consistent only if µ i +μ j ≤ k for any i, j.

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Here * r kills the first column of the lower Young tableau and adds the column of height k −μ 1 to the left-hand side of the upper Young tableau (since µ 1 +μ 1 ≤ k we, thus, get proper Young tableau) and * l acts in the opposite way. Young tableaux that result in consequent application of transformations (A.10) describe one and the same sl(k)-module. Coefficients in (A.8) and (A.9) are chosen such that * r * l T = * l * r T = T .
Quadratic Casimir operator of algebra sl(k) is given by formula For the sl(k)-irrep described by Young tableaux (A.3) C 2 sl(k) is equal to where Σ is the total number of upper indices (total number of cells in the upper Young tableau) andΣ is that of the lower indices. C 2 sl(k) can be also expressed through the heights of the Young tableaux (A.3) Irreducible representations of sl(k)⊕sl(k) are given by tensor product of two sl(k)-irreps is homogeneous polynomial of degree v in two variables Z and D, with coefficients d v s ;j (n a , n b , nā, nb) to be found from the requirement that elements of (B.2) with fixed s span invariant subspace of module M ∞ s , which is submodule M s in decomposition (3.9). Operators (ad ∞ L ) α β , (ad ∞ L )αβ , ad ∞ D , ad ∞ Z (see (3.1)) obviously have the same form in new basis and conserve elements of (B.2), thus, keeping M s invariant. Suppose that operator corresponding to P αβ also keeps M s invariant for some particular choice of g v s . Then as one can easily see it has in basis (B.2) the following form where P v s ;− , P v s ;0 , P v s ;0 , P v s ;+ are some unknown coefficients and Π ⊥ is projector to the traceless component (3.17).

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One can show that solution of system (B.5) is where δ s ;m satisfies the following recurrence equation with boundary conditions δ s ;0 ≡ 1 , δ s ;m<0 ≡ 0. Consider involution of Heisenberg algebra τ : (B.8) It induces involution of algebra su(k, k) τ : As follows from (B.6), (B.7) Euler operators n a , n b , nā , nb contribute to coefficients d v s ;j through the combination n −n only. Therefore elements of (B.2) are invariant up to the factor -1 with respect to τ . Since P αβ and K αβ are τ -conjugated one concludes that K αβ also keeps M s invariant.
The elements of (B.2) are obviously linearly independent and span the whole M ∞ s . Therefore they form the basis of M ∞ s under consideration. Substituting values of P v s ;− , P v s ;0 , P v s ;0 , P v s ;+ found in (B.6) to (B.4) one gets representation of su(k, k) on M s (see (3.16) for exact formulae).
Suppose first that v max = v = 0, i.e. consider terminal termsB t 0 s (0, q, q), q = 0, . . . , s − 1 (see diagram (4.23)). The structure of dotted indices of all such terms is described by the two-row Young tableau with the rows of equal length and, thus, terms (B.12) project it to the two-row Young tableau with the first row less than the second, which is zero. Now lets v max = v > 0, i.e. consider terminal termsB t v s (v, q, q), q = 0, . . . , s − v − 1 in diagram (4.23). As one can easily see from (4.25), (4.26) monomialsm t s −v (v, q, q) corresponding to such terms have value and, thus, Taking into account that operators ∂ 2 ∂a α ∂bβ ,āβ ∂ ∂bα from (B.12) decrease the value of v max by 1 one gets that due to (4.18) elements of (B.12) vanish.

C σ − -cohomology
Let C = {C, ∂} be some co-chain complex. Here C = ⊕ p=0 C p is a graded space and is a differential. The powerful tool to calculate cohomology of C consists in consideration of homotopy operator ∂ *

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According to the standard result of the cohomological algebra (see e.g. [24]) every element of C-cohomology group H has representative belonging to kernel of the anticommutator Θ = {∂, ∂ * } provided that Θ is diagonalizable on C. Indeed, Θ obviously commutes with ∂ and, thus, both operators have common set of eigenvectors. Suppose ψ ∈ C is ∂-closed vector from this set, such that Θψ = qψ, q = 0. Then acting on ψ by operator 1 q Θ one gets that ψ = 1 q ∂∂ * ψ is ∂-exact. In paper [25] it was observed that if along with the above assumptions operator Θ is positive or negative semi-definite, representatives of H are in one-to-one correspondence with the elements from the kernel of Θ.
Let differential ∂ and homotopy ∂ * are given by where P αβ and K αβ are su(k, k) generators of translations and special conformal transformations represented by operators acting in some su(k, k)-module M . Let C be space of differential forms (graded by the rank of the form) taking values in M . In other words C = Λ ⊗ M , where Λ denotes k 2 -dimensional external algebra generated by ξ. Consider subspace Λ p ⊂ Λ of the p-th order monomials in ξ. Algebra sl(k) ⊕ sl(k) is represented on Λ p by operators with quadratic Casimir operator equal to (C.5) In these notation operator Θ has the following form where p is the rank of differential form, D and Z are dilatation and helicity operators represented in M , C 2 u(k,k) is quadratic u(k, k)-Casimir operator ( (C.7) Note that all the ingredients of (C.6) commute and their common eigenvectors form the basis diagonalizing Θ.
For the subsequent analysis we also need to study sl(k) ⊕ sl(k)-tensorial structure of Λ p . To this end consider the basis element of Λ p Ξ(p) α 1α1 ···αpαp = ξ α 1α1 · · · ξ αpαp . (C.8) Since ξ-s anticommute one can easily see that symmetrization of any group of undotted indices of Ξ(p) imply the antisymmetrization of the corresponding group of dotted indices and conversely. Therefore, if undotted indices are projected to obey some symmetry conditions 8 corresponding to Young tableau Y with the rows of lengths λ 1 , . . . , λ k , λ 1 + · · · + λ k = p, dotted indices are automatically projected to obey symmetry conditions corresponding to Young tableau Y T with the columns of heights λ 1 , . . . , λ k . Note that all the rows of Y are, thus, required to be not greater than k since in opposite case antisymmetrization of more than k dotted indices implied. On the other hand projection of undotted indices of Ξ(p) to symmetry conditions corresponding to any Young tableau not longer than k (i.e. such that any of its rows are not longer than k) leads to nonzero result. Let us define an operation of transposition T that maps Young tableau Y with rows λ 1 , . . . , λ k to Young tableau Y T with columns λ 1 , . . . , λ k and let Y T be called the transpose of Y. In these notation decomposition of Ξ(p) into sl(k) ⊕ sl(k)-irreducible components is the following Ξ(p) : where Y p,k denote any Young tableau of the length not longer than k and with the total number of cells equal to p.
Using formulas (A.13) and (A.14) one can calculate Casimir operator ξ C 2 sl(k)⊕sl(k) of the sl(k)⊕sl(k)-representations corresponding to Young tableaux listed in decomposition (C.9). It can be easily seen that the terms of (A.13) depending on the lengths of the rows of Y p,k are cancelled out by the terms of (A.14) depending on the heights of the columns of (Y p,k ) T and one finally gets the same result as in (C.5).

C.1 Gauge sector
Consider co-chain complex C s = (C s , σ − ), where C s = Λ ⊗ M s is the space of differential forms taking values in su(k, k)-adjoint module M s and operator σ − = ξ αβ (ad P ) αβ JHEP12(2016)118 is differential. C s is obtained from above consideration if one sets M , P αβ and K αβ equal to M s , (ad P ) αβ and (ad K ) αβ correspondingly. To coordinate notation let homotopy ∂ * = ∂ ∂ξ αβ (ad K ) αβ be denoted as σ * − in this case. Note that we should also require reality of C s , i.e. ζ(ω s ) = ω s , (C.10) where ω s ∈ C s and ζ is given by (1.11) and (5.2). However one can ignore (C.10) until all σ − -cohomology are found. Indeed, suppose (C.10) is disregarded. If H p s is some σ −cohomology, ζ(H p s ) is also σ − -cohomology since differential σ − is real. So the combinations H p s + ζ(H p s ) give us all real σ − -cohomology. Let C p s denote subspace of p-forms in C s . Consider such a scalar product on C s with respect to which involution (B.8) defined on ξ αβ by plays a role of Hermitian conjugation. Such scalar product is obviously positive definite. 9 Due to (B.9) and (C.11) operators σ − and −σ * − are mutually τ -conjugate and, thus, Θ = {σ − , σ * − } is negative semidefinite. Therefore σ − -cohomology H s coincide with the kernel of operator Θ, which can be found by analyzing those elements of C s that correspond to the maximal eigenvalues of Θ.
Substituting the value of C 2 su(k,k) (see (3.27)) to (C.6) one gets If conformal weight ∆ and differential form rank p are fixed the maximal value of Θ corresponds to the maximum of C 2 sl(k)⊕sl(k) . The general element of C p s has the following form ω p s = ω p s γ 1 ...γp;γ 1 ...γp; β(n b );β(nb) α(na);α(nā) ξ γ 1γ1 · · · ξ γpγp a α(na)āα(nā) b β(n b )bβ (nb) . (C.13) Its sl(k) ⊕ sl(k)-tensorial structure is described by Young tableaux found in tensor product of those describing sl(k) ⊕ sl(k)-structure of Ξ(p) (see (C.4)) and of M s (see (3.15)), i.e. ω p s : where (n) denotes one-row Young tableau of length n. So now we need to find such an irreducible component from tensor product (C.14) that maximizes C 2 sl(k)⊕sl(k) . Let us first consider undotted part of (C.14). Suppose Y p,k = (λ 1 , . . . , λ h ), i.e. consists of h ≤ k rows of lengths k ≥ λ 1 ≥ · · · ≥ λ h > 0, λ 1 + · · · + λ h = p. As one can readily see from formula (A.13) for the value of sl(k)-Casimir operator depending of the rows of Young tableau, the maximal value of sl(k)-Casimir corresponds to the undotted component of (C.14) with the upper row (n a ) symmetrized to the first row of Y p,k (i.e. located in the mostly upper manner) and without any contractions done between Y p,k and the lower row (n b ). The same arguments true for the dotted part of (C.14) also. Let such component be denoted as where the left-hand side (before the comma) corresponds to the undotted part and the righthand side (after the comma) to the dotted part of Young tableau, ⊗ U denotes the mostly upper component in tensor product, ⊗ ⊥ denotes traceless component (i.e. without any contractions done) in tensor product. The letters on the top denote from what constituents (either some oscillators or basis 1-forms ξ) corresponding Young tableau is composed. For instance a (n a ) denotes the row of length n a composed from oscillators a. To calculate the value of C 2 sl(k)⊕sl(k) for component (C. 15) it is convenient to use rowformula (A.13) for its undotted part and column-forma (A.14) for its dotted part. One gets (C.16) As seeing from (C.16) C 2 sl(k)⊕sl(k) does not depend on the shape of Young tableau Y p,k except the length of the first row λ 1 , the number of rows h and the total number of cells p.
Substituting (C. 16) to (C.12) and expressing all the variables in terms of independent coordinates on M s (3.23), (3.24) one gets Due to inequalities q, t ≤ s − v − 1 and k ≥ 2 one can see that the last term in (C.17) is negative for v > 0 and, thus, maximization of Θ requires v = 0. So we finally arrive at Let H p s;∆ denote p-th σ − -cohomology corresponding to the module M s with conformal weight ∆. First consider degenerate case s = 1. Since M 1 is trivial and σ − ≡ 0 one gets that cohomology H p 1,0 are all real p-forms Suppose now that s > 1. All zeros of (C.18) and corresponding σ − -cohomology are listed below.

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1. v = q = t = 0, i.e. n a = nā = 0, n b = nb = s − 1 and ∆ = −s + 1. Formally (C. 18) does not impose any additional limitations on λ 1 and h (recall that we always require λ 1 , h ≤ k), but according to argument given in appendix A (see page 24) traceless tensor identically vanishes if corresponds to Young tableau with total hight of some upper and lower columns greater than k. Since due to (C.15) cohomology in this Analogously to item 1 to get nonzero result we additionally require h ≤ k − 1 in first case and λ 1 ≤ k − 1 in complex conjugated case. We, thus, have We have Substituting k = 2 to above formulae one gets (5.18).

C.2 Weyl sector
Consider co-chain complexC s = (C s ,σ − ), whereC s = Λ ⊗M s is the space of differential forms taking values in su(k, k) twist-adjoint moduleM s andσ − = ξ αβ (tw P ) αβ . Unfortunately the powerful homotopy technic described at the beginning of this section is not applicable in the case under consideration. This is because of the sign change in twist transformation (4.1), which breaks mutual Hermitian conjugacy of (tw P ) αβ and (tw K ) αβ with respect to any positive definite scalar product. Therefore anticommutator ofσ − with homotopyσ * − = ∂ ∂ξ αβ (tw K ) αβ is indefinite. 10 For the purposes of the present paper one need to know 0-th and 1-stσ − -cohomology only and also can fix k = 2. Let us focus on this case leaving general situation for the future investigation.
(C. 23) Recall that (up to an overall factor) the basis inM s isg v sm s−v (n a , nā, n b , nb), v = 0, . . . , s − 1, wherem s−v are monomials of form (4.12) that can be fixed by independent coordinates (v, q, t) (4.27). Since k = 2 one can rewritem s−v as follows where b α = b β αβ and αβ is totally antisymmetric tensor. Such monomials form sl(k)⊕sl(k) irrep corresponding to Young tableau 11 In what follows we denote diagrams like (C.25) as (l 1 , l 2 ), where l 1 and l 2 are the numbers of cells of undotted and dotted rows correspondingly. Decompose operatorσ − into the sum of three operators in accordance with their action on the basis elements ofM sσ where ϕ(n) = 1/(n + 2) andΠ ⊥ is projector 12 given by (4.17). From the nilpotency ofσ − it follows that Representative ofσ − -cohomology can always be chosen to have definite conformal weight and definite irreducible sl(k) ⊕ sl(k)-structure. General element of Λ p ⊗M s ;∆ with 11 This Young tableau is obtained from (4.13) by Hodge conjugation (A.10). 12 ProjectorΠ ⊥ b αm s−v which acts on the oscillator b α with risen index carries out symmetrization of b α with all a-s and b-s inms−vΠ

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fixed conformal weight∆ can be decomposed as where summandF p;v s;∆ = Λ p ⊗g v s (m s−v +m s−v +· · · ) is a linear combination of basis elements with v and∆ fixed and tensored by Λ p .
Within this decompositionσ − -closedness condition forF p s;∆ split into the system . So in order toF p s;∆ be p-thσ − -cohomology one can require its term with the lowest value of v to be p-thσ − − -cohomology. Let us find 1-stσ − − -cohomologyh −;1 . Note that unlike the whole operatorσ − operatorsσ ±,0 − acting separately map monomials into monomials and, thus, one can look for cohomologyh −;1 among irreducible components of tensor product ξ ⊗m s −v . These components are described by Young tableaux obtained in tensor product of (C.25) with one undotted and one dotted cells. Closedness and exactness of each component can be easily checked by direct computation. In (C.31) the results are collected (n a +n b +1, nb −nā +1) , (n a +n b −1, nb −nā −1) , (n a +n b +1, nb −nā −1) , (n a +n b −1, nb −nā +1) , n a +n b ≥ 0 , n a +n b ≥ 1 , n a +n b ≥ 0 , n a +n b ≥ 1 , We thus have two series ofF 1;v min s;∆ that pretend to contribute toH p s;∆ . Let us consider both series separately.
(2) Component (n a −1, nb −nā +1) with n b = 0. In this case coordinates of corresponding 2. their coordinate v is greater than v min but less than v max ; 3. contribute to component (C.36) when tensored by ξ.
Suppose the elementm we are looking for has coordinates (v , q , t ). Let the orders ofm with respect to oscillators (which are expressed via coordinates through the formula (4.28)) be denoted as n a , n b , n ā , n b . Since tensoring by ξ either adds or subtracts one cell to/form Young tableau we require n a + n b , n b − n ā = n a ( − 2), nb − nā [ + 2] , (C.37) where n a , nā , nb are given in (C.35) and numbers in parenthesis (brackets) could be either skipped or taken into account. Condition (C.37) guaranties that ξ ⊗m contains component (C.36). In terms of coordinates the requirements above give the following system