BRST approach to Lagrangian formulation for mixed-symmetry fermionic higher-spin fields

TL;DR

This work develops a BRST/BFV framework to construct gauge-invariant Lagrangians for irreducible half-integer higher-spin fields with two-row Young tableaux in flat space. By embedding the fields into an auxiliary Fock space, identifying a mixed system of first- and second-class constraints, and performing a Verma-module–based auxiliary representation, the authors convert to a first-class constraint system and build a nilpotent BRST operator. The resulting Lagrangian actions describe both massless and massive fermionic higher-spin fields with reducible gauge symmetries, and are illustrated with explicit two-row examples, including spin configurations (3/2,1/2) and (3/2,3/2). The approach provides a universal, off-shell-constraint–free method that reproduces the correct irreducible Poincaré representations through BRST cohomology and paves the way for future AdS/generalized-interaction extensions and higher-row generalizations.

Abstract

We construct a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with the corresponding Young tableaux having two rows, on a basis of the BRST approach. Starting with a description of fermionic higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space, 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 auxiliary representations of the constraint subsuperalgebra containing the subsystem of second-class constraints in terms of Verma modules. We propose a universal procedure of constructing gauge-invariant Lagrangians with reducible gauge symmetries describing the dynamics of both massless and massive fermionic fields of any spin. No off-shell constraints for the fields and gauge parameters are used from the very beginning. 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. To illustrate the general construction, we obtain a Lagrangian description of fermionic fields with generalized spin (3/2,1/2) and (3/2,3/2) on a flat background containing the complete set of auxiliary fields and gauge symmetries.