A note on higher-order vertices of higher-spin fields in flat and (A)dS space

Euihun Joung,Massimo Taronna

doi:10.1007/jhep09(2020)171

Abstract

In this work we classify (homogeneous) solutions to the Noether procedure in (A)dS for an arbitrary number of external legs and in general dimensions, analysing also the corresponding deformations of gauge symmetries. This builds upon the corresponding flat space classification [1], which we review and give its relation with the (A)dS result presented here. The role of dimensional dependent identities is studied in detail, which we find do not lead to new solutions for couplings involving more than three fields. For spins one and two our formalism recovers the Yang-Mills and Gravity examples.

Highlights

HS theories as topological string theories could be a way forward
We first give a self-contained introduction of the main concepts of the Noether procedure program [10], for the sake of brevity leaving out most of the technical details which can be found in the literature
In constructing gauge-invariant interaction vertices, we focus for simplicity on the transverse and traceless (TT) part of the latter disregarding the terms proportional to divergences and traces of the fields

Summary

B H -couplings

Is expressible (up to field redefinitions) as a H-coupling namely a function of the Hij structures (1.13) which are off-shell gauge invariant: see e.g. the diagram A of figure 1. One can view this problem in the opposite perspective, as the question whether there are additional couplings besides those which are trivially gauge-invariant and expressed in terms of the H-structures: see the diagram B of figure 1 The latter viewpoint is natural in the sense that H-couplings always provide consistent interactions while G-couplings are gauge-invariant on-shell and arise only in some special cases as, for instance, the case of three-massless-field interactions. The G-solutions discussed around (1.14) appear only when differential equations which we derive from the gauge invariance condition admit some singular points.7 At cubic order such singular points exist in the case of. For the other cases of three-field interactions and higher-order interactions, this never happens since the field masses (cubic case) or Mandelstam variables prevent the coefficients of the PDE to vanish identically

Organisation of paper

Homogeneous solution in flat-space

Homogeneous solution in flat-space: A review

General n

Combining solutions: more than one massless field

Homogeneous solution in AdS