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

Alexander A. Reshetnyak

TL;DR

The article develops a universal BRST-BFV framework for free half-integer higher-spin fermions with arbitrary index symmetry, realized through a conversion of mixed first-/second-class constraints into a first-class system using an auxiliary $osp(1|2k)$-based generalized Verma module. A complete BRST operator is constructed for the converted constraint algebra, enabling gauge-invariant unconstrained Lagrangians that describe massless and massive fields of any multi-spin $(n_1+1/2,\dots,n_k+1/2)$. The method is demonstrated via explicit Lagrangians for two-row and three-row Young tableaux, including a new unconstrained formulation for the massless $(5/2,3/2)$ spin-tensor and its massive counterpart, with reducible gauge symmetry of finite order. The work provides a systematic, algebraic route to metric-like Lagrangians for arbitrary mixed-symmetry fermionic HS fields, with potential extensions to AdS spaces and to interacting theories.

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).

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

TL;DR

The article develops a universal BRST-BFV framework for free half-integer higher-spin fermions with arbitrary index symmetry, realized through a conversion of mixed first-/second-class constraints into a first-class system using an auxiliary -based generalized Verma module. A complete BRST operator is constructed for the converted constraint algebra, enabling gauge-invariant unconstrained Lagrangians that describe massless and massive fields of any multi-spin . The method is demonstrated via explicit Lagrangians for two-row and three-row Young tableaux, including a new unconstrained formulation for the massless spin-tensor and its massive counterpart, with reducible gauge symmetry of finite order. The work provides a systematic, algebraic route to metric-like Lagrangians for arbitrary mixed-symmetry fermionic HS fields, with potential extensions to AdS spaces and to interacting theories.

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 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 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).

Paper Structure

This paper contains 25 sections, 235 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Derivation of Half-Integer HS Symmetry Superalgebra in $\mathbb{R}^{1,d-1}$
  3. Converted HS Symmetry Superalgebras for YT with $k$ Rows
  4. Auxiliary Representation for $osp(1|2k)$ Superalgebra
  5. Scalar Oscillator Realization of Additional Parts to Constraints
  6. On Auxiliary Representations of Superalgebra $\mathcal{A}^f(Y(k),\mathbb{R}^{1,d-1})$ for Massive HS Fields
  7. BRST--BFV Operator
  8. Unconstrained Gauge-invariant Lagrangians
  9. Examples
  10. Spin $(n_1+\frac{1}{2},n_2+\frac{1}{2})$ Mixed-symmetric Field
  11. Spin $(n_1+\frac{1}{2},n_2+\frac{1}{2},n_3+\frac{1}{2})$ General mixed-symmetric Field
  12. Spin $(\frac{5}{2},\frac{3}{2})$ Mixed-symmetric Massless Spin-tensor
  13. Reducible Gauge Transformations for Gauge Parameters
  14. Gauge Transformations for Fields
  15. Gauge-invariant Unconstrained Lagrangian
  16. ...and 10 more sections