Constructing the Cubic Interaction Vertex of Higher Spin Gauge Fields

TL;DR

This work develops a BRST- and oscillator-based framework to construct cubic interaction vertices for massless higher-spin fields in both flat and AdS space, using the triplet formulation to describe reducible HS representations. By introducing three copies of the HS functional and imposing BRST invariance, the authors derive general constraints on the cubic vertex, while identifying and factoring out fake interactions via field redefinitions. In flat space they define a rich interacting symmetry algebra and solve the BRST equations in a controlled expansion, recovering known constructions such as Koh’s vertex. The AdS generalization shows how curvature corrections modify the algebra and BRST constraints, and the results serve as a groundwork toward a gauge-invariant, potentially holographically tractable HS Lagrangian. The approach highlights the role of an explicit derivative expansion and clarifies the interplay between gauge invariance, field redefinitions, and holographic considerations.

Abstract

We propose a method of construction of a cubic interaction in massless Higher Spin gauge theory both in flat and in AdS space-times of arbitrary dimensions. We consider a triplet formulation of the Higher Spin gauge theory and generalize the Higher Spin symmetry algebra of the free model to the corresponding algebra for the case of cubic interaction. The generators of this new algebra carry indexes which label the three Higher Spin fields involved into the cubic interaction. The method is based on the use of oscillator formalism and on the Becchi-Rouet-Stora-Tyutin (BRST) technique. We derive general conditions on the form of cubic interaction vertex and discuss the ambiguities of the vertex which result from field redefinitions. This method can in principle be applied for constructing the Higher Spin interaction vertex at any order. Our results are a first step towards the construction of a Lagrangian for interacting Higher Spin gauge fields that can be holographically studied.