Abstract
We analyze free conformal higher spin actions and the corresponding wave operators in arbitrary even dimensions and backgrounds. We show that the wave operators do not factorize in general, and identify the Weyl tensor and its derivatives as the obstruction to factorization. We give a manifestly factorized form for them on (A)dS backgrounds for arbitrary spin and on Einstein backgrounds for spin 2. We are also able to fix the conformal wave operator in d=4 for s=3 up to linear order in the Riemann tensor on generic Bach-flat backgrounds.
Highlights
It is important to keep in mind that HS conformal theories are naturally higher derivative theories and for this reason violate unitarity, just as conformal gravity
The crucial difference between spin 2 and HS fields is the explicit appearance of the Weyl tensor within the gauge variation of the generic two derivative operators
With the help of Mathematica we have worked out the explicit form of the unique spin 3 conformal wave operator in d = 4 up to linear terms in the Riemann tensor on Bach-flat backgrounds
Summary
Conformal higher spin fields [8, 13] can be defined at the linear level by demanding the following gauge invariance properties δξ φμ1···μs = ∇(μ1 ξμ2···μs), δα φμ1···μs = g(μ1μ2 αμ3···μs). In the operator notation the gauge invariance properties (2.1) take the form δξφ(x, u) = u · ∇ξ(x, u), δαφ(x, u) = u2α(x, u) From this it follows that a conformal field can be regarded as an equivalence class of standard massless higher spin fields defined on the cone u2 ∼ 0. This observation allows us to use so-called Thomas-D derivatives ∂ˆu in the auxiliary variable u. The conformal wave operator for higher spins does not factorize on generic Einstein spaces, as we shall demonstrate
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