Bosonic Fradkin-Tseytlin equations unfolded. Irreducible case

We factorize 4d Fradkin-Linetsky higher spin conformal algebra by maximal ideal $I^1-\alpha$ and construct irreducible infinite-dimensional modules $M_\alpha$ of 4d conformal algebra that are parameterized by real number $\alpha$. It is shown that independently of $\alpha$ unfolded system of equations corresponding to each $M_\alpha$ describes collection of Fradkin-Tseytlin equations for all spins $s=1,\dots,\infty$ with zero multiplicity.

It was shown that unfolded system constructed with respect to modules M ∞ ,M ∞ is coordinated with decomposition (1.3) and, thus, decomposes into infinitely many copies of unfolded systems corresponding to free conformal equations on spin s field s = 1, 2, . . . (Fradkin-Tseytlin equations). The degeneration of unfolded system considered in [11] is due to non simplicity of algebra iu(2, 2). Really, it contains an infinite chain of ideals (1.4) iu(2, 2) ⊃ I 1 ⊃ I 2 ⊃ · · · ⊃ I m ⊃ · · · , generated by star powers of Z, i.e. ideal I m is spanned by the elements of form (Z * ) m * h(a, b,ā,b). In [11] it was speculated that unfolded system constructed with respect to irreducible algebra isu 0 (2, 2) = iu(2, 2)/I 1 should contain each spin s = 1, 2, . . . in one copy only.
In the present paper we consider this case in more general formulation. Namely one can construct a series of ideals I 1 α of algebra iu(2, 2) that are spanned by the elements of form (Z − α) * h(a, b,ā,b) for some real number α. Quotients iu(2, 2)/I 1 α give rise to the series of irreducible infinite-dimensional algebras isu α (2, 2).
In this paper we briefly discuss the structure of adjoint and twisted-adjoint u(2, 2)-modules on isu α (2, 2). Consider unfolded system of equations corresponding to these modules and show that, as was speculated in [11] it is decomposed into the subsystems corresponding to Fradkin-Tseytlin equations for one copy of every spin s = 1, 2, . . . independently of α.

Adjoint module
Adjoint action of u(2, 2)-generators on iu(2, 2) is given by formulas where n a , n b , nā , nb are Euler operators for corresponding oscillators. Generators (2.1) commute with spin operator which, thus, decomposes the whole u(2, 2)-module into the submodules with the spin s fixed.
To find adjoint representation of u(2, 2) on quotient algebra isu α (2, 2) one should fix some basis on isu α (2, 2) first. The most natural basis is is polynomial of oscillators, which satisfy conditions (1.1) and is traseless, i.e. satisfy relations However, generators of u(2, 2), namely generators of translation and special conformal transformation, are not diagonal with respect to spin value in this basis. Really factorization requirement (Z − α) * h(a , b ,ā ,b) ∼ 0 doesn't commute with spin operator (2.2).
To avoid this inconvenience let us consider more general ansatz for the basis elements is some polynomial on D of the power v with coefficients d v s;j to be found. Direct computation (analogues to that in [11]) brings us to the following result where δ s;m are some functions depending on i(n −n) with n = n a + n b ,n = nā + nb. Functions δ s;m are fixed by the following recurrence equation with boundary conditions δ s;0 ≡ 1 , δ s;m<0 ≡ 0.
In new basis (2.5) operators L α β ,Lαβ, D, Z are given by the same formulae (2.1) while operators P αβ and K αβ have the following new form where Π ⊥ = (Π ⊥ ) 2 is projector to the traceless components (2.4). Operators (2.9) commute with spin operator s, which in new basis has form We, thus, have that all modules M α of u(2, 2) adjoint action on algebra isu α (2, 2) are isomorphic and decompose into direct sum 2)-submodules of M α , which are spanned by basic vectors (2.5) with s fixed.

Twisted-adjoint module
Twisted-adjoint moduleM α can be obtained from adjoint module M α by twist transformation which preserves commutator Under transformation (3.1) elements (2.5) get the following form (a, b,ā,b) , s = 1, 2 . . . , v = 0, 1, . . . , s − 1 , wheref satisfies twisted-traceless conditions coefficientsd v s;j are given by , andδ s;m are some functions depending on i(n −n), where n = n a + n b is the same as in the previous section andn = nā − nb − 2 is twist-transformed operatorn. Functionsδ s;m are fixed by the following recurrence equation with boundary conditionsδ s;0 ≡ 1 ,δ s;m<0 ≡ 0. The structure of moduleM α differs from that of M α . Firstly, elements (3.3) are not linearly independent. Really, due to (3.4) functionf forms two-row Young tableau with respect to dotted indices with fist (second) row of length nb (nā). Thus, operator (bα ∂ ∂āα ) u vanishes onf for u > v max = nb − nā. Therefore, collection of linearly independent elements of (3.3), which form basis inM α is given by formula It is almost obvious that operators (3.10) commute with spin operators and, thus, moduleM α admit decomposition analogous to that of adjoint case (3.11)M α = ⊕ ∞ s=1M s . Though it is worth to mention that for operators of translation and of special conformal transformation some additional analyses is needed. Really, the second terms of these operators, when acting on the elementg vmax sf with the maximal value of v, decrease the value of v max by one but still don't decrease v and, thus, break the limit v ≤ v max . However, it can be shown that coefficients in front of these problematic terms zero out and therefore decomposition (3.11) holds (see [11] for details).
Finally, let us note that we also need complex conjugated twisted-adjoint moduleM α = −M † α , where † is Hermitian involution defined by the following relations on oscillators
System Operator σ transforms (4.6) into For operatorσ one gets complex conjugated analogs of (4.6) and (4.7). Substituting these values in (2.10) one finds that σ +σ maps 0-form under consideration into 2-for taking values in g 0 s f . So it preserves the spin value and, thus, system (4.1) really split into subsystems with fixed spin.
To analyze dynamical content of system (4.1) one should note first that almost all equations of general unfolded system just express higher order fields as derivatives of lower order fields. (In our case the order of fields is given by their conformal weight.) And dynamically nontrivial aspects are hidden in cohomology of the operatorσ − (see [20] for more details)Ĥ p σ − . In our case (4.8)σ − = σ − +σ − +σ − + σ +σ and • 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 − .
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 ǫ αβ , ǫ αβ , ǫαβ, ǫαβ.

Conclusion
Equations (4.10) and gauge transformations (4.11) correspond to spin s Fradkin-Tseytlin equations and gauge transformations written in spinorial form. We, thus, shown that (independently of α) linear unfolded system constructed with the use of adjoint u(2, 2)-module M α and twisted-adjoint u(2, 2)-modulesM α ,M α correspond to the collection of Fradkin-Tseytlin equations for all spins s = 1, 2, . . . in one copy. Algebraically this is equivalent to decomposition formulas (2.11), (3.11), which show that independently of α all adjoint modules M α are isomorphic and the same is true for twisted-adjoint modulesM α . We also found basis (2.5), (3.8) for modules M α ,M α in which this isomorphism become explicit.
However, the whole algebras isu α (2, 2) are not isomorphic. This can be seen if one considers commutator of some its nonbilinear elements. Thus, as follows from our analyses, the difference between these algebras become relevant on the nonlinear level only.
Finally let us note that some other approaches to Fradkin-Tseytlin equations were suggested in papers [21], [22].