Conserved Higher-Spin Charges in $AdS_4$

Gauge invariant conserved conformal currents built from massless fields of all spins in 4d Minkowski space-time and $AdS_4$ are described within the unfolded dynamics approach. The current cohomology associated with non-zero conserved charges is found. The resulting list of charges is shown to match the space of parameters of the conformal higher-spin symmetry algebra in four dimensions.


Introduction
Gauge invariant conserved currents of different spins in 4d Minkowski space were constructed in [1] in terms of generalized higher-spin (HS) Weyl curvatures introduced originally in [2]. The latter describe on-shell nontrivial gauge invariant combinations of derivatives of fields which generalize the spin-one Maxwell tensor and linearized spin-two Weyl tensor. Later conserved HS currents were also considered in [3,4] while nontrivial conserved currents leading to non-zero charges were identified in [5].
In [6] it was shown that global conformal HS symmetries of 4d massless fields of all spins are described by the Weyl algebra A 4 of eight oscillators. Algebras of symmetries of equations of motion of irreducible free fields and supermultiplets were also found in [6] while extensions to higher dimensions were elaborated in [7,8,9,10].
A convenient way to analyze conserved quantities is by using the language of differential forms in the framework of so-called unfolded approach [11] in which all fields and field equations are formulated in terms of differential forms (for a review see e.g. [12]). Closed forms describing the gauge invariant conservation laws in 4d Minkowski space-time in the unfolded approach were found in [4]. In this paper we extend these results to the case of AdS 4 and analyze the current cohomology characterizing nontrivial conserved charges. Namely, in [13] it was shown that the space of closed three-forms which can give rise to conserved charges is far larger than the space of HS conserved charges that can be associated with symmetries of massless fields. Hence, it was conjectured in [13] that most of the found closed three-forms are exact. In this paper, we show that this is indeed true and that the current cohomology matches the anticipated higher-spin global symmetries. We focus on the gauge invariant currents built in terms of generalized Weyl tensors. Note that non-gauge invariant conserved currents built in terms of HS connections are also available [14] in 4d Minkowski space but not yet in AdS 4 .
The extension of the construction of gauge invariant conserved charges to the AdS 4 background performed in this paper may have various applications allowing in particular to compute HS charges carried by HS black-hole solutions [15,16] in the HS theory which requires the AdS 4 background.
The rest of the paper is organized as follows. In Section 2 the description of higher-rank fields in the unfolded approach is briefly recalled. In Section 3 we recall the unfolded form of free 4d HS equations in AdS 4 proposed in [17] and their flat limit. In Section 4 conserved HS currents in the AdS 4 space-time are constructed in terms of covariantly-constant oscillators and De Rham cohomology of gauge-invariant conserved conformal currents built from massless fields of all spins is found. It is shown that resulting nontrivial charges match the space of parameters of the HS symmetry algebra. In Section 5 conserved currents of 4d Minkowski space are reconstructed from the flat limit of those in AdS 4 .

Higher-rank fields
Conformal massless fields of all spins in four dimensions can be described [6] by a rank-one zero-form C(Y |x) where Y A are auxiliary spinor variables (A, B = 1, ..., 4, are Majorana spinor indices). It is convenient to interpret C(Y |x) as a vector in the Fock space F of the algebra of oscillators Y A and Z A that satisfy commutation relations The Fock vacua |0 and 0| are defined to obey In these terms the rank-one equation of [6] takes the form where W (Y, Z|X) satisfies the flatness condition where h AB , f AB and ω A B are components of the one-form connection. The sp(8) flatness conditions are where A, B = 1, . . . 4 label independent solutions normalized so that This normalization guarantees that A A (X) and B B (X) obey canonical commutation relations at any X Thus, in agreement with the results of [6], global conformal HS symmetries form the Lie algebra associated with the Weyl algebra A 4 of four pairs of oscillators. It contains the sp(8) subalgebra of bilinears of oscillators. In fact, the generating elements of the symmetry parameters A A and B B can be interpreted as supergenerators which, together with their anticommutators The rank-r equations for r species of oscillators can be considered analogously with the Fock space realization of the rank-r field |C(Y i |X) Obviously, these equations are invariant under the global symmetry with parameters η(A, B) valued in the Weyl algebra A 4r generated by Let hs(n; C) be the complex Lie superalgebra resulting from A n via the Z 2 graded commu- where homogeneous elements f (Z, Y ) are associated with even and odd elements, . Then the (complexified) rank-r equations are invariant under hs(4r; C). The rank-one HS algebra hs(4; C) belongs to hs(4r; C). Among a number of inequivalent embeddings of hs(4; C) into hs(4r; C), the principal embedding where the element In the 4d Minkowski setup it is convenient to use two-component spinor notation. In these terms, The 4d conformal algebra su(2, 2) ∼ o(4, 2) has connections h αα ′ , ω α β ,ω α ′ β ′ , b and f αα ′ . Extending su(2, 2) to u(2, 2) by adding a central helicity generator with the gauge connectioñ b, the u(2, 2) flatness conditions read as Here the traceless parts ω L α β andω L α ′ β ′ of ω α β andω α ′ β ′ describe the Lorentz connection, while the traces are associated with the gauge fields of dilatation b and helicity generatorb Reduction of the unfolded equations (2.4) to the u(2, 2) ⊂ sp(8) invariant setup for the massless field C(y,ȳ|x) is and D tw c is the conformal covariant derivative The AdS 4 description with the background fields valued in sp(4) ⊂ su(2, 2) results from the Ansatz and Here indices are raised and lowered by the two-component symplectic forms ε αβ and ε α ′ β ′ The rank-one unfolded equation in the AdS space is 3 Rank-two equations and covariant oscillators in AdS 4 In this paper we are specifically interested in the case of rank-two field |J(Y |x) because, as shown in [13], it is associated with 4d conformal conserved currents built from massless fields of all spins. It satisfies the rank-two current equation In the two-component spinor notation (2.18), after the rescaling the oscillator variables y 2 ,ȳ 2 → iy 2 , iȳ 2 and z 2 ,z 2 → iz 2 , iz 2 for the future convenience, this gives Let us look for a three-form ω, closed by virtue of the rank-two equations, in the form where Ω| is a three-form that obeys the equation which, together with the current equation (3.1), implies that ω is closed On-shell closed forms generate conserved charges Conservation means that Q is independent of local variations of the surface such as local variation of the time parameter. Exact ω = dχ do not contribute to Q for solutions of the field equations that decrease fast enough at space infinity. Hence, nontrivial charges Q are associated with the cohomology of currents. Clearly, any three-form ω(η(A, B)) = Ω|η(A, B)|J , where η(A, B) is any function of the oscillators A i A and B A i (2.17) with i = 1, 2, is also closed. Hence, the full space of closed currents is the space of arbitrary functions of the oscillators (2.17). This freedom should encode the freedom in different HS charges. Indeed, as shown in [4], the realization of a rank-two field in terms of bilinears of rank-one fields gives rise to the full list of conformal gauge invariant conserved currents of all spins in four dimensions, which generalize the so-called generalized Bell-Robinson currents constructed by Berends, Burgers and van Dam [1]. However, the freedom in a function of two sets of oscillators A i A and B A i is far larger than that in HS symmetries of rank-one equations, parametrized by a function of the rank-one variables A A and B A (2.10). Hence, in [13] we conjectured that most of the closed forms (3.8) are exact, generating no nontrivial HS charges. The identification of nontrivial conserved charges in the flat space was done in [5] by a different approach. In this paper we extend these results to the AdS 4 geometry using the methods of unfolded dynamics.
Let us pack the oscillators y α i ,ȳ α ′ i , z i α ,z i α ′ into κ n n α , ζ n n α ′ with n = +, − and n = −, + by setting (3.9) One can see that in these terms the nonzero commutation relations are [κ n k β , κ m n α ] = ε n m ε k n ε βα , where indices are raised and lowered by ε n m , ε n m , ε n m and ε n m with Evidently, from (2.3) it follows In these terms, Eq. (3.3) takes the form Analogously, the covariantly constant oscillators A , B (2.17) are packed into τ n n a (κ, ζ|x) , υ n n a ′ (κ, ζ|x) , a = 1, 2; a ′ = 1, 2 (3.14) so that τ m n a (κ, ζ|0) = κ m n α δ α a , Eq. (3.15) guarantees that the oscillators τ (x) and υ(x) satisfy the canonical commutation relations analogous to (3.10) at any x (3.16) This can also be seen in terms of Killing spinors c β (x) and s β ′ (x) of [13], that obey the equations A basic of the space of solutions of this system is formed by four independent pairs of spinors (c a β (x), s a β ′ (x)) and (c a ′ β (x), s a ′ β ′ (x)) labeled by a = 1, 2 and a ′ = 1, 2 and obeying the following initial conditions at x = 0 where 0 denotes any point of space-time. From these conditions it follows that A specific form of the Killing spinors depends on a chosen coordinate system. The covariantly constant oscillators τ, υ are expressed via the Killing spinors as We observe that the operators Using (3.12) and since the last term is symmetric in k and m while because all indices take two values it has to be proportional to ǫ k m . Then for an arbitrary polynomial η m k the form with Ω m k (4.1) is closed provided that J(Y |x) satisfies the current equation (3.2). The central fact of the analysis of the on-shell cohomology is that each of the forms is exact provided that J(Y |x) solves (3.2).
To prove this fact the following simple formulae formulae are useful and, as a consequence of (3.12), Let us prove that ω a − m (4.5) is exact (other cases are analogous). By virtue of (3.19) we obtain Analogously one can show that the both of the forms in The proof for the other forms in (4.12) is analogous. This fact admits the following interpretation. The bilinears in τ a m n and υ b ′ k k form a Lie algebra sp (16) while form a Lie algebra o(8) that commutes with the horizontal h sl 2 (3.21) acting on the hatted indices. For the space of parameters η polynomial in oscillators, factorization of the generators (4.14) allows us to factor out any combination of oscillators with antisymmetrization of a pair of the hatted Latin indices. 1 Since the forms (4.12) and (4.5) that contain antisymmetrized hatted indices are exact, the leftover forms belong to the space of differential forms ω (4.4) with totally symmetrized hatted indices. To describe such forms consider expressions Λ n (η) = Ω n, m η(τ m k a , υ n p a ′ ) (4.15) polynomial in τ m k a and υ n p a ′ . Let P AdS be the space of polynomials (4.15) with symmetrized hatted indices. Clearly, h sl 2 leaves P AdS invariant. Since any h sl 2 −highest vector has the form for some polynomial η, P AdS is a span of vectors Λ n (η) = 1 n! ad n g − − Ω + + η(τ m + a , υ n + a ′ ) , ad x (y) = [x, y] . Note that [f −+ , g m n ] = 0, [f −+ , Ω m n ] = 0. Hence for a given current field J = J s h ,sv such that with an arbitrary functionc + distinguishing between J s h ,sv with different s h . To distinguish between odd and even currents with respect to spinorial variables it is convenient to consider another generation function Using the Taylor expansion f (x + y) = exp(x ∂ ∂y )f (y), Eqs. (4.23), (4.24) give where η ± (a m + a , b n + a ′ |g) = e h ,ev η e h ,ev (a m + a , b n + a ′ )c ± (g), while To make contact with [13], let us reformulate the result in the notations analogous (modulo some rescalings of variables) to those of [13] with ∂ ±α = z 1 α ± z 2 α , y ±α = y 1α ± y 2α etc. In these terms, nontrivial charges are represented by the closed three-forms where J satisfies the current equation (3.2), a ′ = s a ′ α ′ (x)ȳ − α ′ + c a ′ α (x)∂ +α ,ǭ +a ′ = s a ′ γ ′ (x)∂ +γ ′ + c a ′ α (x)y − α , ̺ −a = c a α (x)∂ −α + s a α ′ (x)ȳ + α ′ , ̺ + a = c a α (x)y + α + s a α ′ (x)∂ −α ′ , ̺ −a ′ = s a ′ γ ′ (x)∂ −α ′ + c a ′ α (x)y + α ,̺ + a ′ = s a ′ α ′ (x)ȳ + α ′ + c a ′ α (x)∂ −α .

Minkowski current cohomology
Analogous results for the flat Minkowski space announced in [13] can be obtained as follows.