Conformal higher spin theory

Abstract

We construct gauge theory of interacting symmetric traceless tensors 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 operatorĤ, and the symmetric tensors emerge after expansion of the latter in power series in derivatives. After decomposition in perturbative series around conformally flat pointĤ = 2 with Euclidean metric, the action functional describes conformal higher spin theory. Namely, the linear in fluctuation term cancels, while the one quadratic in fluctuation breaks up as a sum of conformal higher spin theories, the latter being free gauge theories of symmetric traceless tensors of rank s with with the algebra of observables of the quantized point particle. Each global symmetry produces a Noether current constructed out of ψ according to general formula we present in the paper. In the caseĤ = 2 the algebra of observables is an extension of the conformal algebra decomposed w.r.t. its adjoint action as a direct sum of finite-dimensional representations characterized by traceless rectangular two-row Young tableaux. This infinite-dimensional algebra coincides, in d = 3, 4, 6 with conformal higher spin algebras constructed before in terms of even spinor oscillators, which origin is due to twistor reformulation of the dynamics of the massless particle. The construction of the paper may be a starting point for diverse conjectures. First, we discuss the extension of the geometry "quantized point particle + conformal higher spin fields in d dimensions" to the one "tensionless d − 1 brane + higher spin massless fields in d + 1 dimensions", where the phase of ψ appear to describe transverse motion of the brane inside d + 1-bulk. This picture arises in the semiclassical approximation to the quantized particle dynamics, the latter approximation provides d-dimensional generalization of W -geometry elaborated on by Hull in d = 1, 2 case. Next, 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 conforms with recent proposal (for d = 4) by Tseytlin and provides information on the full higher spin action in AdS d+1 .

One-loop effective actions and higher spins. Part II

“…A particularly interesting one is that of [24] (although the main focus of [24] is in conformal higher spin theories). To this purpose the author utilizes a quantum mechanical approach of a particle coupled to external symmetric fields, that makes essential use of Weyl quantization…”

“…Other topics, such as the discussion of the ambiguities inherent in the choice of the conserved currents in the initial matter model, will be included in a subsequent article. This is a good point to mention that our research is indebted to several preexisting works, in particular with [17][18][19] as far as the inspiration is concerned, with [20][21][22][23][24][25][26][27][28][29] as far as the methods are concerned and with [4,5,[30][31][32][33][34][35][36][37][38][39][40][41] for HS theories. Other papers of ours, related to the present one are, beside [1,42,43].…”

“…It allows also to compute the relevant quadratic effective action of the external fields and, at least in principle, the effective action to all orders. As said above, the interest of [24] is limited to conformal higher spin theories. Later on Bekaert et al [25] have revisited and extended this method, but always considering a massless scalar.…”

“…As compared to [24,25], our method is based on Feynman diagrams together with a considerable technical improvement in computing the latter. In particular, in the present draft, we have extended our previous results in [1] to mixed spin terms, which also turn out to be always made up of a nonlocal transverse part and a local non-transverse one.…”

“…In particular, in the present draft, we have extended our previous results in [1] to mixed spin terms, which also turn out to be always made up of a nonlocal transverse part and a local non-transverse one. The main difference with the methods of [24,25], is that we deal with massive scalar and fermion theories coupled to external sources. The introduction of a mass is important because it allows us to intercept not only the conformal higher spin kinetic terms which can be found in [66], but also the Fronsdal equations for symmetric fields which appear in [4,5].…”