Massive Fields of Arbitrary Integer Spin in Symmetrical Einstein Space

TL;DR

This work tackles the consistency problem of describing gauge fields with arbitrary integer spin in curved backgrounds, specifically symmetrical Einstein spaces, by developing an algebraic, gauge-invariant operator framework in a pseudo-Hilbert space. The authors formulate a Lagrangian as an expectation value ⟨Φ^s|ℒ(L)|Φ^s⟩ and construct the spin-s operator set {L_1,L_{-1},L_2,L_{-2},L_0} so that gauge invariance is encoded in their commutators, reproducing the Fronsdal structure in flat space and accommodating massive states via an extra scalar oscillator with L_1 = p·a + m b, L_2 = 1/2 a·a + b^2, L_0 = p^2 + m^2. Extending to a symmetrical Einstein background, the paper derives curvature-dependent deformations L_1^{(1)} and L_2^{(1)} that restore the essential algebra to linear order in the Riemann tensor R_{μν αβ} and scalar curvature R, yielding gauge-invariant massive-vector and massive-spin-2 Lagrangians in curved space. The results clarify how curvature constraints and mass interact to preserve correct degrees of freedom and causality, offering a systematic route to propagate higher-spin fields in nontrivial geometries and informing potential applications in string theory and holography. Overall, the paper provides a practical, algebraic framework for extending higher-spin gauge theories from flat to curved backgrounds with controlled linear curvature corrections.

Abstract

We study the propagation of gauge fields with arbitrary integer spins in the symmetrical Einstein space of any dimensionality. We reduce the problem of obtaining a gauge-invariant Lagrangian of integer spin fields in such background to an purely algebraic problem of finding a set of operators with certain features using the representation of high-spin fields in the form of some vectors of pseudo-Hilbert space. We consider such construction in the linear order in the Riemann tensor and scalar curvature and also present an explicit form of interaction Lagrangians and gauge transformations for massive particles with spins 1 and 2 in terms of symmetrical tensor fields.