On Lagrangian formulations for mixed-symmetry HS fields on AdS spaces within BFV-BRST approach

The key aspects of a gauge-invariant Lagrangian description of massive and massless half-integer higher-spin fields in AdS spaces with a two-row Young tableaux Y (s1, s2) are presented in an unconstrained description, as well as in off-shell formulations with algebraic constraints, on the basis of BFV-BRST operators for non-linear operator superalgebras, encoding the initial conditions realized by constraints in a Fock space and extracting the higher-spin fields from unitary representations of the AdS group. Problems of a unified description of the known interactions and the variety of elementary particles are revealed at high energies (partially accessible in LHC), thus providing the relevance of the development of higher-spin (HS) field theory due to its close relation to superstring theory in constant-curvature spaces, which operates with an infinite set of bosonic and fermionic HS fields subject to a multi-row Young tableaux (YT) Y (s1, ..., sk), k≥1 (for a review, see [1]). This report deals with the recent results in constructing Lagrangian formulations (LFs) for free fermionic HS fields in AdSd-spaces with Y (s1, s2) in a Fronsdal metric-like formalism within the BFV-BRST approach [4], as a starting tool for a description of interacting HS fields in the framework of quantum field theory, and is partially based on the results presented in [2, 3]. This method of constructing an LF for HS fields, originally developed as applied to a Hamiltonian quantization of gauge theories with a given LF, consists in a solution of the problem inverse to that of the method [4] (as in the case of string field theory [5] and the first papers on HS fields [6]) in the sense of constructing a gauge LF w.r.t. a nilpotentBFV-BRST operator Q. Q is constructed from a system Oα of 1st-class constraints, including a special non-linear operator superalgebra {OI}:{OI} ⊃ {Oα}, defined on an auxiliary Fock space and encoding the relations that extract the fields with a fixed (m, s) from the AdS-group representation spaces. A massive representation with spin s = (s1, s2), si = ni + 1 2 , n1 ≥ n2, of the AdS group in an AdSd space is characterized by Y (s1, s2) and realized in the space of mixed-symmetry spin-tensors Φ(μ)n1 ,(ν)n2 ≡ Φμ1...μn1 ,ν1...νn2A(x) ←→ μ1 μ2 · · · · · · · · · μn1 ν1 ν2 · · · · · · · νn2 , (1) subject to the equations (for β = (2; 3) ⇐⇒ (n1 > n2;n1 = n2), r being the inverse squared AdSd-radius, with a suppressed Dirac index A and the matrices {γμ, γν} = 2gμν(x)),