General Lagrangian Formulation for Higher Spin Fields with Arbitrary Index Symmetry. I. Bosonic fields

TL;DR

This work develops a universal BRST-BFV framework to formulate gauge-invariant Lagrangians for irreducible integer higher-spin bosonic fields with arbitrary index symmetry encoded by Young tableaux with k rows. By embedding fields in an auxiliary Fock space and performing a controlled conversion of second-class to first-class constraints via an auxiliary $sp(2k)$-based Verma-module realization, the authors construct a nilpotent BRST operator $Q$ and an unconstrained action $\mathcal{S}_{(s)_k} = \int d\eta_0 \, {}_{(s)_k}\langle \chi^0 |K_{(s)_k} Q_{(s)_k}| \chi^0 \rangle_{(s)_k}$ that describes both massless and massive bosonic HS fields. The framework automatically yields reducible gauge systems and auxiliary Stueckelberg fields, with the BRST cohomology at ghost number zero reproducing the irreducible Poincaré representations. Explicit constructions are given for two-row and three-row tableaux, including a new unconstrained Lagrangian for the massless spin $(2,1,1)$ field, illustrating the method's power to handle arbitrary index symmetry and higher reducibility levels. This approach significantly extends metric-like formulations of mixed-symmetry HS fields and provides a path toward component Lagrangians and AdS generalizations.

Abstract

We construct a Lagrangian description of irreducible integer higher-spin representations of the Poincare group with an arbitrary Young tableaux having k rows, on a basis of the universal BRST approach. Starting with a description of bosonic mixed-symmetry higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space associated with special Poincare module, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincare-group representations) into a first-class constraint system. For this purpose, we find, for the first time, auxiliary representations of the constraint subalgebra, to be isomorphic due to Howe duality to sp(2k) algebra, and containing the subsystem of second-class constraints in terms of new oscillator variables. We propose a universal procedure of constructing unconstrained gauge-invariant Lagrangians with reducible gauge symmetries describing the dynamics of both massless and massive bosonic fields of any spin. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by the constraints corresponding to an irreducible Poincare-group representation. As examples of the general procedure, we formulate the method of Lagrangian construction for bosonic fields subject to arbitrary Young tableaux having 3 rows and derive the gauge-invariant Lagrangian for new model of massless rank-4 tensor field with spin $(2,1,1)$ and second-stage reducible gauge symmetries.