Bosonic Fradkin-Tseytlin equations unfolded

We test infinite-dimensional extension of algebra su(k,k) proposed by Fradkin and Linetsky as the candidate for conformal higher spin algebra. Adjoint and twisted-adjoint representations of su(k,k) 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.

To construct an infinite-dimensional extension of su(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 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.
The general element X iu(k,k) (a, b,ā,b) of iu(k, k) satisfies the relations [Z, X iu(k,k) ] * = i 2 (n a − nā − n b + nb)X iu(k,k) = 0 , (1. 15) ζ(X iu(k,k) (a, b,ā,b)) =X iu(k,k) (ā,b, a, b) = X iu(k,k) (a, b,ā,b) . (1.16) Picking out contractions a · b andā ·b from X iu(k,k) and taking into account that a · b = D − iZ, a ·b = D + iZ one gets where f u,v is centralized by Z (1.15), satisfies reality condition (1.16) and is traceless with respect to a , b andā ,b, i.e. (1.18) Also note that due to (1.15) Algebra iu(k, k) admits decomposition analogous to (1.14) of u(k, k) Here the infinite sum of u(1) is spanned by the star product powers of Z (1.20) where dots on the right-hand side denote the terms with the lower powers of Z. Therefore general element of subalgebra isu ∞ (k, k) is given by (1.17) with nonconstant f u,0 . Unlike the finite-dimensional case algebra isu ∞ (k, k) is not semi-simple and contains an infinite chain of ideals I m (1.21) isu ∞ (k, k) ⊃ I 1 ⊃ I 2 ⊃ · · · ⊃ I m ⊃ · · · .
Here ideal I m is spanned by the elements of the form where dots on the right-hand side denote the terms of the lower powers of Z and/or D. Let quotient algebras isu ∞ (k, k)/I m be denoted as isu m−1 (k, k) (1.23) isu 0 (k, k) ⊂ isu 1 (k, k) ⊂ · · · ⊂ isu m (k, k) ⊂ · · · ⊂ isu ∞ .
The representatives of the elements of quotient algebra isu m (k, k) can be chosen to have form (1.17) with f u,v ≡ 0 for u > m. Algebra isu 0 (k, k), spanned by elements independent of Z, is semi-simple. In what follows we omit index 0 and denote it isu(k, k). Let us note that in paper [2] algebras isu ∞ (2, 2) and isu (2,2) were denoted as hsc ∞ (4) and hsc(4) correspondingly, where hsc means higher spin conformal and 4 indicates that they extend 4-dimensional conformal algebra. In the later paper [5] these algebras were denoted as cu(1, 0|8) and hu 0 (1, 0|8), where 8 indicates the number of oscillators used and pair 1,0 points out that the above algebras have trivial structure in spin 1 Yang-Mills sector.
The rest of the paper is organized as follows. In section 2 we recall some relevant facts about unfolded formulation. The structure of adjoint representations of algebra su(k, k) on the vector spaces of isu m (k, k) m = 0, 1, . . . , ∞ is discussed in section 3. In section 4 we study twisted-adjoint representation of su(k, k). In section 5 unfolded formulation of 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 basis where adjoint and twisted-adjoint modules from sections 3 and 4 are decomposed 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 first order differential equations [7] (see [8] for a review) 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 form of rank p Ω + 1. It is composed by virtue of exterior product 2 of the elements of W Ω (x), which are contracted with constant functions ..ΨΦ... . Compatibility conditions of (2.1) require F Ω (W ) to satisfy identities In terms of constants f Ω Φ 1 ...Φn conditions (2.4) have a form of generalized Jacobi identities where left-hand side of (2.5) is (anti)symmetrised according to (2.3). Any solution of (2.5) defines a free differential algebra (FDA) [9]. 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 (2.6) δW , where ǫ Ω (x) are p Ω − 1-form gauge parameters. 2 In this paper all products of differential forms are supposed to be exterior and we omit the designation of exterior product in formulae. 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 (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 Let us treat all other fields from W Ω (x) as fluctuations of W A (x). For the sake of simplicity we 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 [10]). System (2.1) linearized over W A (x) reduces to As one can readily see compatibility conditions (2.5) require matrices (T A ) a b and (T A ) i j to form some representations of Lie algebra g. Let corresponding modules be denoted as M andM . Then D and D from (2.10) and (2.11) define M andM-covariant derivatives respectively. Both derivatives are nilpotent (2.12) D 2 = 0 ,D 2 = 0 as a consequence of the zero curvature condition (2.8).
As it was argued in [10] 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) j , right-hand side of (2.10) can be removed by field redefinition And conversely if some field redefinition removing 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 C i (x) of zero curvature condition (2.8) is fixed, the above gauge symmetry breaks down to the global symmetry that keeps W A 0 stable. Parameter of this symmetry ǫ A 0 (x) should obviously satisfy equation 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. The only thing we require is that component of W A 0 corresponding to generator of generalized translation (i.e. generalized coframe) is of maximal possible rank. Let us consider system (2.10), (2.11) As follows from the above consideration it is globally g-invariant with respect to transformations (2.19), (2.20) with W A = W A 0 substituted. This system is also gauge invariant with respect to gauge transformations where ǫ a (x) is 0-form gauge parameter associated with field ω a (x).
To analyze dynamical content of system (2.23), (2.24) let us first consider the case when right-hand side of (2.23) equals zero. In this case equations (2.23), (2.24) are independent and both have a form of covariant constancy conditions. Suppose that modules M andM are graded and this grading is bounded from below. Decompose covariant derivatives (2.23), (2.24) 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 grading was considered in [12]) 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 n-th level of M. Analyzing equation (2.23) and its gauge symmetries (2.25) level by level starting from the lowest grading one can see [13] that those fields which are not σ − -closed (they are called auxiliary fields) expressed by (2.23) as derivatives of lower level fields, where 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 (2.25). Leftover fields (that are called dynamical fields) belong to H 1 σ − the 1-st cohomology of σ − . We also get that differential gauge parameters (i.e. those that does not correspond to Stueckelberg gauge symmetry) belong to H 0 σ − . Let E n denote the left-hand side of (2.23) on 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 considered at 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 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 (2.23) and H 2 σ − are isomorphic as h-modules.
Summarizing, the dynamical content of equation (2.23) with 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 dynamical fields respectively. Analogously for equation (2.24) the dynamical fields and differential equations correspond toH 0 σ − andH 1 σ − . To analyze system (2.23), (2.24) with nonzero right-hand side let us consider operator System (2.23), (2.24) can be rewritten in the following form where new fieldΨ is a pairΨ = (ω, C) incorporating 1-forms ω and 0-forms C and operators (2.29) are extended by zero on the space where they undefined. OperatorD 0 is nilpotent due to compatibility conditions of system (2.23), (2.24). Gauge transformations (2.25) takes a form (2.31) δΨ =D 0Υ , whereΥ = (ǫ, 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 and C p−1 are correspondingly p-form taking values in the module M and p − 1form taking values in the 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 for some elements ω p−1 , C p−2 . Let p-thσ − -cohomology be defined as quotient Since the above analysis of equation (2.23) with zero right-hand side is based on Bianchi identities 4 (2.27) only it is applicable to equation (2.30). We therefore obtain that dynamical content of system (2.30) (or equivalently of system (2.23), (2.24) with nonzero right-hand side) is defined byĤσ − . Namely: • differential gauge parameters are given byĤ 0 σ − ; • dynamical fields are given byĤ 1 σ − ; • differential equations on dynamical fields are in one-to-one correspondence withĤ 2 σ − . In section 5 we use the above technic to analyze dynamical content of su(2, 2)-invariant unfolded system that was originally introduced in [6]. 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 isu ∞ (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 this su(k, k)-module. Obviously, operators commute with adjoint action of su(k, k) (3.1). Moreover due to centralization requirement (1.15) M ∞ s ⊃ I 1 s ⊃ · · · ⊃ I s−1 s , 4 In fact we also required D to have unique σ − and grading to be bounded from below which is obviously also true for (2.30).
where I m is isu ∞ (k, k)-ideal (1.22) and In what follows we omit index 0 and denote M 0 , M 0 s as M, M s , respectively.
As shown in Appendix B.1 module M ∞ s admits the following decomposition . . , ∞ , 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 the 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 and forms eigenvector corresponding to eigenvalue s ′ − v with respect to operators s 1 and s 2 (3.2) (3.14) In other words f s ′ −v is a sum of monomials of the form ;β(nb) ;α(nā) is a traceless complex tensor 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 (3.15) should be coordinated with s ′ and v through formula (3.14).
Let In what follows we study the structure of module M s ′ and in particular show that it is irreducible.
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 up to an overall factor Z s−s ′ by the following operators where n = n a + n b ,n = nā + nb, ϕ(n) = 1/(n + k) and Π ⊥ = (Π ⊥ ) 2 is the projector to the traceless component (3.13) . Elements of the form (3.20) and (3.21) with v = 0 have the lowest conformal weight −s ′ + 1 and the highest conformal weight s ′ − 1 correspondingly.
All elements B v s ′ can be arranged on the following diagram Here every dot (•) indicates some B v s ′ . All dots in the same row correspond to B v 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 diagram. The lowest (highest) dot in each rhombus corresponds to the B v s ′ min (B v s ′ max ) (see (3.20), (3.21)) of the lowest (highest) conformal weight for given v. Arrows indicate transformations which change orders of B v s ′ -s with respect to oscillators a, b,ā,b in such a way that s ′ is kept constant and ∆ increases by 1. Namely It is convenient to introduce independent 5 "coordinates" on diagram (3.22) Here v numerates the rhombus in (3.22) 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. For instance coordinates Let us note that complex conjugation (1. 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 (1.16) coordinate- Let m ′ and m ′′ denote some sl(k) ⊕ sl(k)irreducible submodules of l ′ s ′ and l ′′ s ′ correspondingly. Since all B v s ′ -s in (3.22) with fixed conformal weight (i.e. those contained in fixed row of (3.22)) have different sl(k) ⊕ sl(k)-structure, one necessarily concludes that m ′ and m ′′ coincide with some B v s ′ of (3.22). On the other hand as one can easily see from (3.17) the only B v s ′ from (3.22) annihilated by (ad P ) αβ is B 0 s ′ min given by formula (3.20) with v = 0. We thus conclude that module M s is irreducible and is generated from B 0 s ′ min by (ad K ) αβ .
From (3.17) 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) onM ∞ can be obtained from that of M ∞ by simple replacement (4.1). We have (cf. (3.17)) where nb is the Euler operator for oscillatorb. We requireM ∞ to be annihilated by tw ∞ Analogously to M ∞ moduleM ∞ can be decomposed into direct sum of submodulesM ∞ s picked out by requirement 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 ∞ Here is annihilated by twist-adjoint action of su(k, k) (4.3), In other wordsf s ′ −v can be represented as a sum of monomials ; ;β(nb),α(nā) is a complex traceless tensor separately symmetric with respect to upper and lower group of undotted indices and of the symmetry type described by two-row 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 the formula (4.10).
Letg v s ′ (Z,D)m s ′ −v be denoted asB v s ′ . Due to the above argumentsB v s ′ forms, with respect to generators (tw ∞ L ) α β , (tw ∞ L )αβ, irreducible sl(k) ⊕ sl(k)-module corresponding to the Young tableau (see Appendix A for more details). Analogously to adjoint case letĨ m denote submodule ofM ∞ spanned by the elements (4.7) with the power ofZ grater or equal to m and letĨ m s =Ĩ m ∩M s denote corresponding submodule ofM s . Note thatĨ m s ≡ 0 for m ≥ s. Let us define quotient modules In what follows we omit index 0 and denoteM 0 andM 0 s as M andM s , respectively.
To find how P αβ and K αβ act on (4.7) one should apply twist transformation to (3.17), (3.18). We have Although all the above formulae were obtained by application of twist transformation (4.1) (which respect commutators (4.2)) to analogous formulae corresponding to adjoint module the structure of twist-adjoint modules and its analysis have some important nuances in comparison with adjoint case.
Firstly, twist-adjoint modules are infinite-dimensional. This is because operators 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 adjoint case the elementsB v s ′ of twist-adjoint module are not linearly independent. Indeed, as was discussed abovem s ′ −v forms with respect to dotted indices sl(k)-module corresponding to two-row Young tableaux with first row of length nb and second row of length nā (see (4.13) and Appendix A). Therefore where (· · · ) are some coefficients. In what follows let elementsB v s ′ and corresponding monomials m s ′ −v with v = v max be called terminal and denote them asB t v s ′ andm t s ′ −v correspondingly.
Taking the above arguments into account one concludes that for fixed s ′ all linearly independent B v s ′ -s can be arranged in the following diagram . . .
Here every dot (•) indicates someB v s ′ . All dots in the same row correspond toB v s ′ − s with the same conformal weight (indicated on the left axis) to infinity. Dots compose the collection of strips of width (number of dots) s ′ − v and of infinite length, which are distinguished by the value of v = 0, 1, . . . , s ′ − 1 indicated on the bottom of diagram. The lowest dot in each strip corresponds to theB v s ′ min (4.23) of the lowest conformal weight for given v. Arrows indicate transformations which change orders ofB v s ′ -s with respect to oscillators a, b,ā,b in such a way that s ′ is kept constant and∆ increases by 1. Namely (4.24) ✒ : n a → n a + 1 , Introduce independent coordinates on (4.21) 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 (4.26)  21). Looking on the right-hand sides of (4.16) one sees that their 1-st, 3-d and 4-s terms also conserve (4.21), but 2-nd terms once act at the terminal termB t v s ′ maps it toB ′v+1 s ′ , which due to (4.20) is equivalent to the sum of terms corresponding to s ′ − 1 , s ′ − 2 , . . . So it looks likeM s is not su(k, k)-invariant. In Appendix B.2 it is shown that this is not the case, since these problem terms are zero out either (for v = v max > 0) because coefficients in (4.20) vanish or (for v = v max = 0) because projector (4.17) gives zero.
We, thus, show that analogously to adjoint case modulesM ∞ s ,M m s andM m admit decompositions . . , ∞ and the basis ofM ∞ s that respects these decompositions is whereB v s ′ -s are listed in diagram (4.21). In the same manner as for module M s one can show thatM s is irreducible with quadratic Casimir operator given by formula (3.28).
In what follows we also need complex conjugate modulesM n s , n = 0, 1, . . . , ∞ that are obtained fromM n s by operation of complex conjugation
Let us now consider how operator σ ∞ acts on the basis elements (4.28) of the moduleM ∞ s . Suppose first that s ′ = s − 1 and, thus, the power ofZ in (4.28) is 1. It is easy to see that From (5.10) one gets that for general s ′ and analogous formulae for operatorσ ∞ acting on basis ofM ∞ s . Therefore after field redefinition Here ω s ′ , C s ′ ,C s ′ stand for fields taking values in corresponding modules and To find σ let us consider how operator σ ∞ acts on the elementsB v s ′ =g v s ′ms ′ −v , which form the basis ofM s ′ . Since σ ∞ setsb = 0, onlyb independent components ofB v s ′ are valid. This means that all oscillatorsb inm v s ′ should be transformed 6 byg v s ′ into oscillatorsā. Therefore to give nonvanishing resultm v s ′ is required: firstly, to be of the order not higher than v with respect tob and, secondly, to be independent ofā. From the diagram (4.21) one finds that the onlyB v s ′ -s satisfying to these conditions are those with the lowest conformal weightB v s ′ min (see (4.23)), which correspond to the lowest dot of the strip v in the diagram (4.21). From (4.23) taking into account (B.21) one gets that Here we exploited that requirementā = 0 in (5.18) sets to zero all nonminimal terms, requirement b =ā along with multiplication by v! is analogous to (ā ∂ ∂b ) v in (5.17) and, finally, that v = nā after we setā =b. The form of operatorσ can be obtained from (5.18) by complex conjugation.
As one can see from (5.18) σ maps 7 minimal elementB v s ′ min ofM s ′ corresponding to the stripe v in the diagram (4.21) into the element of M s ′ of conformal weight v that corresponds to the upper left edge of the rhombus 0 in the diagram (3.22), whileσ maps complex conjugate elementB v s ′ min of M s ′ into the complex conjugate element of M s ′ corresponding to the upper right edge of rhombus 0.
As was discussed in section 2 the dynamical content of system (5.15) is encoded by cohomology of operators σ − ,σ − andσ − . Let H p s ′ ;∆ ,H p s ′ ;∆ andH p s ′ ;∆ denote p-th cohomology of corresponding operator with conformal weight ∆. These cohomology are found in Appendix C. For cohomology of 6 Recall thatg v s ′ is a polynomial ofZ andD, which contain operatorsā · ∂ ∂b . 7 To be precise due to reality conditions imposed on M s ′ operator σ does not map into M s ′ , but the sum σ +σ does.
and tensors ε , ϕ , E and S are traceless. For cohomology ofσ − we have . where tensors C ,Ẽ ,S are traceless and symmetry type ofẼ (S) with respect to undotted indices corresponds to two row Young tableaux with first row of length s ′ − 1 and second row of length 1 (2) (see Appendix A for more details). Cohomology ofσ − are complex conjugate to (5.21).

Conclusion
We have proposed unfolded system (5.15) 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 representations on space of algebra isu m (2, 2), m = 0, 1, . . . , ∞ can be decomposed into independent subsystems of form (5.15) by means of appropriate field redefinition and found spectrum of spins for any isu m (2, 2). In particular we have shown that system of equations proposed in [6] (5.6), (5.7) describes linear dynamics of conformal fields of all integer spins greater or equal than 1, where each spin enters 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 higher spin theories is a higher spin algebra. Our results pretend to be a probe of different candidates to this role. We see that algebras isu m (2, 2), m = 1, 2 . . . mediate between isu ∞ (2, 2) and isu (2,2).
Having in mind that conformal higher spin theory has to be somehow related to AdS higher spin theory one can speculate 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 [6], which correspond to isu ∞ (2, 2), are considerably simpler than (5.15) corresponding to isu(2, 2). Therefore an interesting question arises wether it is possible to simplify (5.15) maybe by mixing again gauge and Weyl sectors of the theory (5.14). Another important area of investigation is to consider super extensions of isu m (2, 2) and, thus, bring fermions into the play.

Acknowledgement
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.
Algebra sl(k) ⊕ sl(k) ⊂ u(k, k) is generated by L α β andLαβ (see (1.9)). Let first summand of sl(k) ⊕ sl(k) that is generated by L α β be referred to as undotted sl(k) and second summand that is generated byLαβ be referred to as dotted sl(k). All finite-dimensional 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 β 1 (λ 1 ),...,β k (λ k ) , λ 1 ≥ · · · ≥ λ k ∈ Z + ,λ 1 ≥ · · · ≥λ k ∈ Z + , written in symmetric basis or equivalently by sl(k)-tensors  . 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 8 be denoted as Y[µ 1 , . . . , µ h ]. Using totally antisymmetric tensors ǫ α[k] , ǫ α[k] one can rise and lower indices of T ...,β j [μ j ],... . Taking into account that tensor product of two ǫ-tensors can be written as 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 the 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. Due to the same arguments operations of rising and of lowering indices (A.4), (A.5) result in a tensor satisfying irreducibility conditions only when applied to the highest (i.e. first) column of T . Let define rising and lowering Hodge conjugations by formulas * r (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) and Σ is that of the lower indices. C 2 sl(k) can be also expressed through the heights of the Young tableaux (A.3) (A.14) The irreducible representations of sl(k) ⊕ sl(k) are given by tensor product of two irreps of sl(k) where first sl(k) acts on undotted indices and second sl(k) acts on dotted indices. Tensors T are supposed to satisfy irreducibility conditions above separately within undotted and dotted indices. This representation can be described by four Young tableaux the submodule M s ′ enters to M ∞ s with the factor Z s−s ′ . Therefore the natural ansatz for such a basis is Here f s ′ −v (a, b,ā,b) is traceless (i.e. satisfies (1.18)) and forms eigenvector corresponding to eigenvalue s ′ − v with respect to operators s 1 and s 2 and 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 value of s ′ span invariant subspace of module M ∞ s , which is submodule M s ′ in decomposition (3.9). Operators (3.1) corresponding to the su(k, k) elements L α β ,Lαβ , D , Z obviously conserve elements of (B.2) and a fortiori keep 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 (3.18).
The requirement that (ad ∞ P ) αβ keeps M s ′ invariant reduces to the following system of recurrence equations Here n = n a + n b ,n = nā + nb and ϕ(n) = 1/(n + k). One can show that the solution of system (B.5) is where δ s ′ ;j satisfies the following recurrence equation with boundary conditions δ v s ′ ;v ≡ 1 , δ v s ′ ;j>v ≡ 0. Consider involution (i.e. involutive antilinear antiautomorphism) of Heisenberg algebra It induces involution of the algebra su(k, k) 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 involutive transformation τ (B.8). 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 a 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 the representation of su(k, k) on M s (see (3.17) for exact formulas).
Let us note that Z s−s ′ g v s ′ satisfy to reality conditions (1.11), i.e. (B.10) Few lower examples of g v s ′ are the following Hered v s ′ ;j are coefficients satisfying to twisted equations (B.5). These equations have the following solution (cf. (B.6), (B.7)).
, whereδ s ′ ;j satisfies the recurrence equation with boundary conditionsδ v s ′ ;v ≡ 1 ,δ v s ′ ;j>v ≡ 0. Recall that n = n a + n b andn = nā − nb − k. Now according to arguments given in page 16 we need to show that 2-nd terms of right-hand sides of (4.16) vanish when acting on terminal elementB t v s ′ , i.e.
Suppose first that v max = v = 0, i.e.B t 0 s ′ corresponds to the very left dots of the stripe indicated by v = 0 in the diagram (4.21). As one can easily see the structure of dotted indices of all suchB t 0 s ′ -s is described by two-row Young tableau with rows of equal length and, thus, terms (B.17) project it to two-row Young tableau with first row less than second, which is zero. Now lets v max = v > 0, i.e.B t v s ′ corresponds to the very left dots of the stripe indicated by v in the diagram (4.21). As one can easily see from (4.23), (4.24) the monomialsm t s ′ −v corresponding to such terms have value and operators ∂ 2 ∂a α ∂bβ ,āβ ∂ ∂bα from (B.17) decrease it by 2. Let us find the form of functiong v s ′ in (B.17). Substituting the value of n −n to (B.16) one finds that and, thus, Taking into account that operators ∂ 2 ∂a α ∂bβ ,āβ ∂ ∂bα from (B.17) decrease the value of v max = v by 1 one gets (B.17).
Finally let us note that formula (B.11) in twist-adjoint case is Appendix C. σ − -cohomology Let C = {C, ∂} be some co-chain complex. Here C = ⊕ p=0 C p is graded space and (C.1) ∂ : C p → C p+1 , ∂ 2 = 0 is differential. The powerful tool to calculate cohomology of C consists in consideration of homotopy operator ∂ * According to standard result of cohomological algebra (see e.g. [14]) 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, they have the 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 [15] 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.
Consider subspace Ξ p ⊂ Ξ of the p-th order monomials in ξ. Algebra sl(k) ⊕ sl(k) is represented on Ξ p by operators with quadratic Casimir operator equal to 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 ( Note that all the ingredients of (C.6) commute and their common eigenvectors form basis diagonalizing Θ.
For subsequent analysis we also need to study sl(k) ⊕ sl(k)-tensorial structure of Ξ p . To this end consider the basis element of Ξ p Since ξ-s anticommute one can easily see that symmetrization of any group of undotted indices of Ξ(p) imply antisymmetrization of corresponding group of dotted indices and conversely antisymmetrization of any group of dotted indices imply symmetrization of corresponding group of undotted indices. Therefore, if undotted indices are projected to obey some symmetry conditions 9 corresponding to Young tableau Y with 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 columns of heights λ 1 , . . . , λ k . Note that all rows of Y are 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 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 the sl(k) ⊕ sl(k)-irreducible components can be written as follows (C.9) Ξ(p) : where Y p,k denote any Young tableau of 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 terms of (A.13) depending on lengths of rows of Y p,k are cancelled out by terms of (A.14) depending on heights of 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 = Ξ ⊗ 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 is differential. C s is obtained from the 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.
However one can ignore (C.10) until all σ − -cohomology are found. Indeed, suppose (C.10) disregarded. If H p s is some σ − -cohomology, ζ(H p s ) is also σ − -cohomology since differential σ − is real. So 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 τ (B.8) redefined on ξ αβ by (C. 12) τ : ξ αβ ↔ ∂ ∂ξ αβ plays a role of Hermitian conjugation. Such scalar product is obviously positive definite 10 .
Due to (B.9) and (C.12) 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 maximal eigenvalues of Θ.
(C.15) ω p s : where (n) denotes one-row Young tableau of length n. So now we need to find such an irreducible component of the tensor product (C.15) which maximizes C 2 sl(k)⊕sl(k) . Let us first consider undotted part of (C.15). 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 the 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 undotted component of (C.15) 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.15) also.
Let such component be denoted as where left-hand side (before the comma) corresponds to undotted part and right-hand side (after the comma) to 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 a. To calculate the value of C 2 sl(k)⊕sl(k) for the component (C.16) it is convenient to use row-formula (A.13) for its undotted part and column-forma (A.14) for its dotted part. One gets (C.17) As seeing from (C.17) C 2 sl(k)⊕sl(k) does not depend on the shape of Young tableau Y p,k except the length of first row λ 1 , the number of rows h and the total number of cells p.
Substituting (C.17) to (C.13) and expressing all variables in terms of independent coordinates on M s (3.24), (3.25) one gets Due to inequalities q, t ≤ s − v − 1 and k ≥ 2 one sees that the last term in (C.18) is non positive and, thus, maximization of Θ requires v = 0. 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 Suppose now that s > 1. All zeros of (C.19) and corresponding σ − -cohomology are listed below.
C.2. Weyl sector. Consider co-chain complexC s = (C s ,σ − ), whereC s = Ξ ⊗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 this case. 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 11 . 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.
where b α = b β ǫ αβ and ǫ αβ is totally antisymmetric tensor. Such monomial forms sl(k) ⊕ sl(k) irrep corresponding to the Young tableau 12 In what follows we denote diagrams like (C.26) as (l 1 , l 2 ), where l 1 and l 2 are the number of cells of undotted and dotted rows correspondingly. Decompose operatorσ − into the sum of three operators in accordance with their action on basis elementB v s . Namely operatorσ − − (σ + − ) decrease (increase) v by 1 andσ 0 − does not change it where ϕ(n) = 1/(n + 2) andΠ ⊥ are projectors 13 given by (4.17). From the nilpotency ofσ − it follows that where summandF p;v s;∆ = Ξ p ⊗g v s (m s−v +m ′ s−v + · · · ) is a linear combination of basis elements with fixed v and∆ tensored by Ξ p and v vary from v min to v max .
Within this decompositionσ − -closedness condition forF p s;∆ split into the system                               σ (C.30) 12 This Young tableau is obtained from (4.13) by Hodge conjugation (A.10). 13 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 inm s−vΠ According to the first equation of (C.30)F . Thenσ − -exact shiftF p s;∆ −σ − ε p−1,v min +1 s;∆+1 zeros out termF p;v min s;∆ . So in order forF p s;∆ to 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 components of tensor product ξ⊗m s ′ −v . These components are described by Young tableaux obtained in tensor product of (C.26) with one undotted and one dotted cells (C.31) (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 , Closedness and exactness of each component can be easily checked by direct computation. In (C.31) the results are collected. We thus have two series ofF 1;v min s;∆ that pretend to contribute toH p s;∆ . Let us consider both series separately.
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 (4.26)) 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 (C.37) (n ′ a + n ′ b , n ′ b − n ′ā ) = (n a + n b ( − 2), nb − nā [ + 2]) , where n a , n b , 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 above requirements give the following system where q min = s − v min + m − 1 and t min = s − v min − 1 are corresponding coordinates ofm min . Looking at the diagram (4.21) one sees that if we take into account both numbers (in parenthesis and brackets), v min increases by 2 and, thus, the width of the stripe decreases by 2 while t min decreases by 1 only. Since monomialm min is at the boundary of the stripe v min monomialm ′ is out of the boundary of stripe v min + 2, i.e. does not belong toM s . The same situation occurs if one takes into account the number in parenthesis only. So the only possibility is (C.40) v ′ = v min + 1 , q ′ = q min , t ′ = t min − 1 v min < s − 1 , n ′ a = n a , n ′ b = 0 , n ′ā = nā − 1 , n ′ b = nb + 1 . By means of analogous analysis checking ε 0 s;s+2 -s that could have contributed toF 1 s;s+1 one can show thatF 1 s;s+1 is notσ − -exact. Therefore (C.44)H 1 s;s+1 =g s−1 s ξ γβx α(s) ;β(s) ǫ αγ a α(s+1)bβ(s−1) .