  Conformal Higher Spin Gravity is a higher spin extension of Weyl gravity and
is a family of local higher spin theories, which was put forward by Segal and
Tseytlin. We propose a manifestly covariant and coordinate-independent action
for these theories. The result is based on an interplay between higher spin
symmetries and deformation quantization: a locally equivalent but manifestly
background-independent reformulation, known as the parent system, of the
off-shell multiplet of conformal higher spin fields (Fradkin-Tseytlin fields)
can be interpreted in terms of Fedosov deformation quantization of the
underlying cotangent bundle. This brings into the game the invariant quantum
trace, induced by the Feigin-Felder-Shoikhet cocycle of Weyl algebra, which
extends Segal's action into a gauge invariant and globally well-defined action
functional on the space of configurations of the parent system. The same action
can be understood within the worldline approach as a correlation function in
the topological quantum mechanics on the circle.
