The σ− Cohomology Analysis for Symmetric Higher-Spin Fields

Alexey S. Bychkov,Kirill A. Ushakov,Mikhail A. Vasiliev

doi:10.3390/sym13081498

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 σ− cohomology both in the tensor language in Minkowski space of any dimension and in terms of spinors in AdS4. In the d-dimensional case Hp(σ−) is computed for any p and interpretation of Hp(σ−) is given both for the original Fronsdal system and for the associated systems of higher form fields.

Highlights

Higher-spin (HS) gauge theory is based on works of Fronsdal [1] and Fang and Fronsdal [2], where the action and equations of motion for massless gauge fields of any spin were originally obtained in flat four-dimensional Minkowski space
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
Free unfolded equations for massless HS fields are analyzed in detail in terms of σ− cohomology

Summary

Introduction

Higher-spin (HS) gauge theory is based on works of Fronsdal [1] and Fang and Fronsdal [2], where the action and equations of motion for massless gauge fields of any spin were originally obtained in flat four-dimensional Minkowski space. (For a review see [7].) The no-go theorems implied the existence of the s = 2 barrier suggesting that the construction of an interacting local HS theory in Minkowski space-time is impossible. The proof of these theorems essentially uses the specific form of the algebra of isometries of Minkowski space. The s = 2 barrier in flat space can be overcome in the space-time with non-zero sectional curvature, for example, in the anti-de Sitter space [8] In these spaces it becomes possible to formulate a consistent nonlinear theory of fields of all spins [9,10]. One of the essential features of this approach, which is very useful for analysing symmetries of a given system, is that the variables in the equations are valued in one or another representation of the underlying symmetry algebra

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