Nonlinear Operator Superalgebras and BFV-BRST Operators for Lagrangian Description of Mixed-symmetry HS Fields in AdS Spaces

TL;DR

The work develops a BFV-BRST framework for free mixed-symmetry higher-spin fields in $AdS_d$ with two-row Young tableaux $Y(s_1,s_2)$, encoding AdS irrep constraints into nonlinear (super)algebras $ cal{A}$ and $ cal{A}_b$. It introduces an additive conversion to first-class constraints via Verma-module realizations in an auxiliary Fock space, enabling construction of an exact nilpotent BRST operator $Q'$ with terms up to cubic in ghosts. This BRST operator yields gauge-invariant Lagrangian formulations for HS fields with mass parameter $m$ and spins $ extbf{s}=(s_1,s_2)$, including off-shell variants and a reducible gauge structure. The results provide a systematic route to unconstrained and constrained Lagrangian descriptions of AdS HS fields, tying together nonlinear algebra, Verma modules, and BRST quantization. Open issues remain, notably explicit Verma-module constructions for the full converted superalgebras and extending the approach to fermionic cases and constrained LFs.

Abstract

We study the properties of nonlinear superalgebras $\mathcal{A}$ and algebras $\mathcal{A}_b$ arising from a one-to-one correspondence between the sets of relations that extract AdS-group irreducible representations $D(E_0,s_1,s_2)$ in AdS$_d$-spaces and the sets of operators that form $\mathcal{A}$ and $\mathcal{A}_b$, respectively, for fermionic, $s_i=n_i+{1/2}$, and bosonic, $s_i=n_i$, $n_i \in \mathbb{N}_0$, $i=1,2$, HS fields characterized by a Young tableaux with two rows. We consider a method of constructing the Verma modules $V_\mathcal{A}$, $V_{\mathcal{A}_b}$ for $\mathcal{A}$, $\mathcal{A}_b$ and establish a possibility of their Fock-space realizations in terms of formal power series in oscillator operators which serve to realize an additive conversion of the above (super)algebra ($\mathcal{A}$) $\mathcal{A}_b$, containing a set of 2nd-class constraints, into a converted (super)algebra $\mathcal{A}_{b{}c}$ = $\mathcal{A}_{b}$ + $\mathcal{A}'_b$ ($\mathcal{A}_c$ = $\mathcal{A}$ + $\mathcal{A}'$), containing a set of 1st-class constraints only. For the algebra $\mathcal{A}_{b{}c}$, we construct an exact nilpotent BFV--BRST operator $Q'$ having nonvanishing terms of 3rd degree in the powers of ghost coordinates and use $Q'$ to construct a gauge-invariant Lagrangian formulation (LF) for HS fields with a given mass $m$ (energy $E_0(m)$) and generalized spin $\mathbf{s}$=$(s_1,s_2)$. LFs with off-shell algebraic constraints are examined as well.