Conformal Higher Spin Theory

TL;DR

This paper develops a gauge theory framework for interacting conformal higher spin fields in various dimensions, generalizing previous models, and explores their symmetries, geometric origins, and potential holographic relations to AdS higher spin theories.

Contribution

It constructs a comprehensive conformal higher spin gauge theory in any dimension, extending known free theories to include interactions and exploring their symmetries and geometric interpretations.

Findings

Constructed a gauge theory of interacting symmetric traceless tensor fields of all ranks.

Identified infinite-dimensional conformal higher spin algebras in any dimension.

Proposed a holographic conjecture relating conformal higher spin actions to AdS higher spin actions.

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}.