Towards Lagrangian dynamics for constrained mixed-symmetric interacting higher-spin fields

TL;DR

This work develops a consistent deformation framework for irreducible interacting mixed-symmetric massless higher-spin fields on flat Minkowski space using an incomplete BRST operator $Q_c$ together with holonomic traceless and Young-symmetry constraints. It shows that, beyond imposing these constraints, the deformation must preserve a closed Abelian superalgebra involving $Q_c$, the spin operators, and the holonomic constraints, and requires BRST-closed, traceless, and Young-symmetric cubic vertices, constructed explicitly via traceless and Young-symmetry projectors. The paper provides local, BRST-closed cubic vertices that maintain the same number of physical degrees of freedom as the free theory and outlines how these results relate to the complete BRST formulation and Labastida constrained form. The framework generalizes to higher-order vertices (quartic and beyond) while preserving consistency of constrained higher-spin dynamics, with potential implications for higher-spin interactions in flat or curved backgrounds.

Abstract

The necessary and sufficient conditions to construct consistent Lagrangian formulation for irreducible interacting massless higher-spin (HS) fields on $d$-dimensional Minkowski space within approach with incomplete BRST operator and off-shell holonomic constraints are found. It is shown that in addition to superconmmuting of incomplete BRST operator with appropriate traceless and Young constraints, which annihilate the field and gauge parameter vectors, these constraints should form Abelian superalgebra both with BRST operator and with operators of cubic, quartic and etc. vertices. The consistent deformation of free model with constrained HS fields with integer spin requires for the cubic vertex to be by BRST-closed, traceless and Young-symmetric solution of the generating equations. The explicit form for the vertices for irreducible constrained interacting fields are obtained by means of projectors on traceless and Young-symmetric modes.