Frame- and Metric-Like Higher-Spin Fermions

Rakibur Rahman

doi:10.3390/universe4020034

Abstract

Conventional descriptions of higher-spin fermionic gauge fields appear in two varieties: the Aragone–Deser–Vasiliev frame-like formulation and the Fang–Fronsdal metric-like formulation. We review, clarify and elaborate on some essential features of these two. For frame-like free fermions in Anti-de Sitter space, one can present a gauge-invariant Lagrangian description such that the constraints on the field and the gauge parameters mimic their flat-space counterparts. This simplifies the explicit demonstration of the equivalence of the two formulations at the free level. We comment on the subtleties that may arise in an interacting theory.

Highlights

Arbitrary-spin massless particles are expected to play a crucial role in the understanding of quantum gravity
It is believed that the tensionless limit of string theory is a theory of higher-spin gauge fields
The study of fermionic fields is interesting in this regard because they are required by supersymmetry

Summary

Introduction

Arbitrary-spin massless particles are expected to play a crucial role in the understanding of quantum gravity. A crucial property of frame-like fermions in flat space is their shift symmetry w.r.t. a gauge parameter, which is an irreducible tensor-spinor in the fiber space with the symmetry property of the Young diagram Y(n − 1, 1) This symmetry makes it almost manifest that the free Lagrangian is equivalent to that of the metric-like formulation [1]. In AdS space, the constraints on this parameter may receive nontrivial corrections, which vanish in the flat limit [39,40] This is tantamount to having no such corrections provided that some appropriate mass-like terms appear in the gauge transformation. A review of frame-like higher-spin massless fermions in flat space appears, where we write down the free Lagrangian [40,42] and discuss its gauge symmetries along with the constraints on the field and the gauge parameters.

Frame-Like Fermions in Flat Space

Frame-Like Fermions in AdS Space

Equivalence of Frame- and Metric-Like Formulations

Remarks