Gauge invariants and Killing tensors in higher-spin gauge theories
Xavier Bekaert, Nicolas Boulanger
TL;DR
The paper addresses the problem of classifying gauge-invariant local functions and Killing tensor fields for free completely symmetric higher-spin fields in Minkowski space, comparing constrained and unconstrained metric-like formulations. It develops a BRST-based, cohomological framework on jet space, showing that gauge-invariant functions are generated by the curvature $R$ (unconstrained) or by the Fronsdal tensor $F$ together with the traceless curvature and double trace (constrained), and that on-shell these invariants reduce to functions of the traceless Weyl-like tensor $W$. It then analyzes local Koszul–Tate cohomology to characterize Killing tensors, proving an isomorphism between off-shell Killing tensors and the symmetric algebra of the Poincaré algebra modulo trace relations, and identifying on-shell Killing tensors with the Minkowski higher-spin algebras $hs_\infty$ and $hs$; this provides a direct link between Killing tensors and higher-spin symmetry in flat space and aligns with the unfolded HS framework. The results unify BRST cohomology, Killing-tensor structures, and higher-spin algebras, offering a solid algebraic foundation for classifying local invariants and guiding the construction of interactions in flat-space higher-spin theories.
Abstract
In free completely symmetric tensor gauge field theories on Minkowski space-time, all gauge invariant functions and Killing tensor fields are computed, both on-shell and off-shell. These problems are addressed in the metric-like formalisms.