General Lagrangian Formulation for Higher Spin Fields with Arbitrary Index Symmetry. 2. Fermionic fields

TL;DR

This paper develops a universal BRST-based Lagrangian framework for describing arbitrary higher-spin fermionic fields with mixed symmetry, including both massless and massive cases, using advanced algebraic methods.

Contribution

It introduces a new method for constructing unconstrained, gauge-invariant Lagrangians for fermionic higher-spin fields with arbitrary Young tableaux, extending previous bosonic approaches.

Findings

Derived gauge-invariant Lagrangian for massless rank-3 spin-tensor (5/2,3/2).

Obtained non-gauge Lagrangian for massive rank-3 spin-tensor (5/2,3/2).

Established a universal procedure for arbitrary mixed-symmetry fermionic fields.

Abstract

We continue the construction of a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with an arbitrary Young tableaux having $k$ rows, on a basis of the BRST--BFV approach suggested for bosonic fields in our first article (Nucl. Phys. B862 (2012) 270, [arXiv:1110.5044[hep-th]). Starting from a description of fermionic mixed-symmetry higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space associated with a 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 system of first-class constraints. To do this, we find, in first time, by means of generalized Verma module the auxiliary representations of the constraint subsuperalgebra, to be isomorphic due to Howe duality to $osp(k|2k)$ superalgebra, and containing the subsystem of second-class constraints in terms of new oscillator variables. We suggest a universal procedure of finding unconstrained gauge-invariant Lagrangians with reducible gauge symmetries, describing the dynamics of both massless and massive fermionic fields of any spin. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by constraints corresponding to an irreducible Poincare-group representation. As examples of the general approach, we propose a method of Lagrangian construction for fermionic fields subject to an arbitrary Young tableaux having 3 rows, and obtain a gauge-invariant Lagrangian for a new model of a massless rank-3 spin-tensor field of spin (5/2,3/2) with first-stage reducible gauge symmetries and a non-gauge Lagrangian for a massive rank-3 spin-tensor field of spin (5/2,3/2).