Unconstrained Higher Spins of Mixed Symmetry. II. Fermi Fields

TL;DR

This work delivers a comprehensive local metric-like formulation for free mixed-symmetry fermionic higher-spin fields in flat space by extending Labastida's constrained framework to unconstrained Lagrangians. It develops two unconstrained Lagrangian families (a minimal higher-derivative version and a non-minimal one with only one derivative) and derives the corresponding field equations from Bianchi identities and self-adjointness, including a full treatment of two-family and general N-family cases. The authors classify Weyl-like symmetries, provide explicit reducible (3,2) examples, and extend the construction to irreducible and multi-form fermions, with a systematic procedure to eliminate higher derivatives. They also discuss DOF counts, current exchanges, and the implications for string spectra and possible extensions to interactions, AdS backgrounds, and supersymmetric extensions. Overall, the paper establishes a robust, self-consistent framework for unconstrained higher-spin fermions that connects to geometry, gauge structure, and potential string-theoretic applications.

Abstract

This paper is a sequel of arXiv:0810.4350 [hep-th], and is also devoted to the local "metric-like" unconstrained Lagrangians and field equations for higher-spin fields of mixed symmetry in flat space. Here we complete the previous constrained on-shell formulation of Labastida for Fermi fields, deriving the corresponding constrained Lagrangians both via the Bianchi identities and via the requirement of self-adjointness. We also describe two types of unconstrained Lagrangian formulations: a "minimal" one, containing higher derivatives of the compensator fields, and another non-minimal one, containing only one-derivative terms. We identify classes of these systems that are invariant under Weyl-like symmetry transformations.