Conformal Higher Spin Theory
A. Y. Segal
TL;DR
The paper constructs a bosonic conformal higher spin theory in $d>2$ as a gauge theory of symmetric traceless tensors of all ranks, with an action given by ${ m A}[H]={ m Tr}\, heta(H^*)$, where $H$ is a Hermitian differential operator encoding the HS background. Perturbing around a conformally flat vacuum $H= abla^2$, the quadratic action decomposes into a direct sum of free conformal HS theories across all spins, revealing an infinite-dimensional conformal HS algebra ${ m chs}(d-q,q)$. The work develops a geometric picture in which HS fields arise as background fields of a quantum point particle, constructs a full invariant induced action with a semiclassical low-energy expansion, and discusses connections to AdS/CFT via boundary HS data and a tensionless-brane interpretation. It also introduces dressing/undressing maps to relate deformed and undeformed HS theories, analyzes Noether currents for a scalar coupled to HS backgrounds, and situates the construction within the broader Fradkin-Linetsky/AdS HS framework. The results provide a consistent nonlinear framework for interacting conformal HS fields and illuminate holographic and geometric facets of HS gauge symmetry.
Abstract
We construct gauge theory of interacting symmetric traceless tensor fields of all ranks s=0,1,2,3, ... which generalizes Weyl-invariant dilaton gravity to the higher spin case, in any dimension d>2. The action is given by the trace of the projector to the subspace with positive eigenvalues of an arbitrary hermitian differential operator H, the symmetric tensor fields emerge after expansion of the latter in power series in derivatives. After decomposition in perturbative series around a conformally flat point H=\Box, the quadratic part of the action breaks up as a sum of free gauge theories of symmetric traceless tensors of rank s with actions of d-4+2s order in derivatives introduced in 4d case by Fradkin and Tseytlin and studied at the cubic order level by Fradkin and Linetsky. Higher orders in interaction are well-defined. We discuss in detail global symmetries of the model which give rise to infinite dimensional conformal higher spin algebras for any d. We stress geometric origin of conformal higher spin fields as background fields of a quantum point particle, and make the conjecture generalizing this geometry to the system "tensionless d-1 brane + Fronsdal higher spin massless fields in d+1 dimensions". We propose a candidate on the role of Higgs-like higher spin compensator able to spontaneously break higher spin symmetries. At last, we make the conjecture that, in even dimensions d, the action of conformal higher spin theory equals the logarithmically divergent term of the action of massless higher spin fields on AdS_{d+1} evaluated on the solutions of Dirichlet-like problem, where conformal higher spin fields are boundary values of massless higher spin fields on AdS_{d+1}, the latter conjecture provides information on the full higher spin action in AdS_{d+1}.