The $\sigma_-$ Cohomology Analysis for Symmetric Higher-Spin Fields

TL;DR

This paper rigorously proves the First On-Shell Theorem for symmetric massless higher-spin fields by calculating the $\sigma_-$ cohomology in various formalisms, clarifying the dynamical structure of these fields in different dimensions.

Contribution

It provides a complete proof of the First On-Shell Theorem for arbitrary spin fields using $\sigma_-$ cohomology calculations in tensor and spinor formalisms.

Findings

Cohomology $H^p(\sigma_-)$ computed for all $p$ in any dimension.

Interpretation of cohomology results for Fronsdal and higher form systems.

Unified treatment of higher-spin fields in Minkowski and $AdS_4$ spaces.

Abstract

In this paper, we present a complete proof of the so-called First On-Shell Theorem that determines dynamical content of the unfolded equations for free symmetric massless fields of arbitrary integer spin in any dimension and arbitrary integer or half-integer spin in four dimensions. This is achieved by calculation of the respective $\sigma_-$ cohomology both in the tensor language in Minkowski space of any dimension and in terms of spinors in $AdS_4$. In the $d$-dimensional case $H^p(\sigma_-)$ is computed for any $p$ and interpretation of $H^p(\sigma_-)$ is given both for the original Fronsdal system and for the associated systems of higher form fields.