Real Forms of Complex Higher Spin Field Equations and New Exact Solutions

TL;DR

The work extends four-dimensional higher-spin gauge theory to spacetimes with signatures $(4-p,p)$ and nonzero cosmological constant, incorporating Euclidean and Kleinian chiral sectors via spinor-oscillator realizations of Vasiliev-type equations. It develops the complex field equations, derives five real-forms, and analyzes a chiral limit, then constructs four classes of exact solutions (Types 0–3) using a gauge-function method, including a Type 3 construction with all HS fields nonzero. The Type 1 solutions yield $SO(4-p,p)$-invariant deformations with a continuous parameter and discrete projector data, while Type 2 and Type 3 reveal nontrivial internal master-field data and higher-spin generalizations of Type D gravitational instantons, representing the first known exact 4D HS backgrounds with massless HS excitations. Together, these results provide explicit backgrounds to study HS interactions, spectra, and geometric features in non-Lorentzian signatures, and motivate future work on HS geometry, invariants, and quantum-gravity applications.

Abstract

We formulate four dimensional higher spin gauge theories in spacetimes with signature (4-p,p) and nonvanishing cosmological constant. Among them are chiral models in Euclidean (4,0) and Kleinian (2,2) signature involving half-flat gauge fields. Apart from the maximally symmetric solutions, including de Sitter spacetime, we find: (a) SO(4-p,p) invariant deformations, depending on a continuous and infinitely many discrete parameters, including a degenerate metric of rank one; (b) non-maximally symmetric solutions with vanishing Weyl tensors and higher spin gauge fields, that differ from the maximally symmetric solutions in the auxiliary field sector; and (c) solutions of the chiral models furnishing higher spin generalizations of Type D gravitational instantons, with an infinite tower of Weyl tensors proportional to totally symmetric products of two principal spinors. These are apparently the first exact 4D solutions with non-vanishing massless higher spin fields.