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 operator H , and the symmetric tensors emerge after expansion of the latter in power series in derivatives. After decomposition in perturbative series around conformally flat point H=□ 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 actions of d−4+2 s order in derivatives (in odd dimensions they are boundary terms), for all integer s, 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. The action appears to be the unique functional invariant w.r.t. general similarity transformations H′=e ω ̂ † He ω ̂ , the latter invariance plays the role of gauge symmetry group of the model. In the framework of the perturbative decomposition, the Hermitian part of ω gauges away the trace parts of the symmetric tensors parameterizing the fluctuation, while the anti-Hermitian one provides standard linearized gauge transformations of conformal higher spin fields. The action can be calculated as a semiclassical series in ℏ which counts the number of space–time derivatives and thereby exhibits itself as a parameter of low-energy expansion, like α′ in string theory, in so doing the classical term is given by the volume of the domain H( x, p)>0 (where H( x, p) is the Weyl symbol of H ), it does not contain derivatives and is interpreted as a cosmological term. At the same time, further terms of the ℏ-expansion are given by integrals of distributions localized on the constraint surface H( x, p)=0, and the conformal higher spin- s action arises from the ℏ d−4+2 s -correction. The full gauge invariance of the model is interpreted as covariance algebra of generalized Klein–Gordon equation H|ψ〉=0 for complex scalar field ψ.